The heat capacity of the gas. The heat capacity of the body ST is the ratio of the amount of heat Q communicated to the body to the change in temperature ∆T

Heat capacity body (usually denoted by the Latin letter C) - physical quantity determined by the ratio of an infinitesimal amount of heat δ Q received by the body to the corresponding increment of its temperature δ T :

$C = (\delta Q \over \delta T).$

The unit of heat capacity in the International System of Units (SI) is J / .

Specific heat

Specific heat capacity is the heat capacity per unit quantity of a substance. The amount of a substance can be measured in kilograms, cubic meters and moles. Depending on which quantitative unit the heat capacity belongs to, there are mass, volume and molar heat capacity.

Mass specific heat capacity ( With), also called simply specific heat capacity, is the amount of heat that must be supplied to a unit mass of a substance in order to heat it by a unit temperature. In SI, it is measured in joules per kilogram per kelvin (J kg −1 K −1).

And at constant pressure

$c_p = c_v + R = \frac(i+2)(2) R.$

The transition of a substance from one state of aggregation to another is accompanied by spasmodic a change in heat capacity at a specific temperature point of transformation for each substance - the melting point (transition of a solid into a liquid), the boiling point (transition of a liquid into a gas) and, accordingly, the temperatures of reverse transformations: freezing and condensation.

The specific heat capacities of many substances are given in reference books, usually for a process at constant pressure. For example, the specific heat capacity of liquid water under normal conditions is 4200 J / (kg K); ice - 2100 J/(kg K).

Heat capacity theory

There are several theories of the heat capacity of a solid:

The Dulong-Petit law and the Joule-Koppe law. Both laws are derived from classical concepts and are valid with a certain accuracy only for normal temperatures (approximately from 15 °C to 100 °C).
Einstein's quantum theory of heat capacities. The first application of quantum laws to the description of heat capacity.
Quantum theory of Debye's heat capacities. Contains the most complete description and agrees well with experiment.

The heat capacity of a system of non-interacting particles (for example, an ideal gas) is determined by the number of degrees of freedom of the particles.

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Notes

Literature

// Encyclopedic Dictionary of a Young Physicist / V. A. Chuyanov (ed.). - M .: Pedagogy, 1984. - S. 268–269. - 352 p.

An excerpt characterizing heat capacity

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Material from the Uncyclopedia

The heat capacity of a body is the amount of heat that must be imparted to a given body in order to raise its temperature by one degree. When cooled by one degree, the body gives off the same amount of heat. The heat capacity is proportional to the mass of the body. The heat capacity of a unit mass of a body is called specific, and the product of specific heat capacity by atomic or molecular weight is called atomic or molar, respectively.

The heat capacities of different substances vary greatly. So, the specific heat capacity of water at 20 ° C is 4200 J / kg K, pine wood - 1700, air - 1010. For metals, it is less: aluminum - 880 J / kg K, iron - 460, copper - 385, lead - 130. The specific heat increases slightly with temperature (at 90°C, the heat capacity of water is 4220 J/kg K) and changes strongly during phase transformations: the heat capacity of ice at 0°C is 2 times less than that of water; the heat capacity of water vapor at 100°C is about 1500 J/kg K.

The heat capacity depends on the conditions in which the temperature of the body changes. If the dimensions of the body do not change, then all the heat goes to change the internal energy. Here we are talking about heat capacity at constant volume (C V). At constant external pressure, due to thermal expansion, mechanical work is performed against external forces, and heating to a particular temperature requires more heat. Therefore, the heat capacity at constant pressure C P is always greater than C V . For ideal gases C P - C V \u003d R (see figure), where R is the gas constant, equal to 8.32 J / mol K.

Usually measured C P . The classical method for measuring heat capacity is as follows: a body whose heat capacity (C x) they want to measure is heated to a certain temperature t x and placed in a calorimeter with an initial temperature t 0 filled with water or another liquid with a known heat capacity (C k and C w are the heat capacities of the calorimeter and liquids). By measuring the temperature in the calorimeter after thermal equilibrium (t) has been established, the heat capacity of the body can be calculated using the formula:

C x \u003d (t-t 0) (C f m f + C to m k) / (m x (t x -t)),

where m x , m w and m k are the masses of the body, liquid and calorimeter.

The most developed theory is the heat capacity of gases. At ordinary temperatures, heating leads mainly to a change in the energy of the translational and rotational motion of gas molecules. For the molar heat capacity of monatomic gases C V theory gives 3R/2, diatomic and polyatomic - 5R/2 and 3R. At very low temperatures, the heat capacity is somewhat less due to quantum effects (see quantum mechanics). At high temperatures, vibrational energy is added, and the heat capacity of polyatomic gases increases with increasing temperature.

The atomic heat capacity of crystals, according to the classical theory, is equal to 3Ry, which is consistent with the empirical law of Dulong and Petit (established in 1819 by French scientists P. Dulong and A. Petit). The quantum theory of heat capacity leads to the same conclusion at high temperatures, but predicts a decrease in heat capacity with decreasing temperature. Near absolute zero, the heat capacity of all bodies tends to zero (the third law of thermodynamics).

Ways to change the internal energy of the body

There are two ways to change the internal energy of a body (system) - doing work on it or transferring heat. The process of exchange of internal energies of contacting bodies, which is not accompanied by the performance of work, is called heat transfer. The energy that is transferred to the body as a result of heat transfer is called the amount of heat received by the body. The amount of heat is usually denoted by Q. Generally speaking, a change in the internal energy of a body in the heat transfer procedure is the result of the work of external forces, but this is not work associated with a change in the external parameters of the system. This is the work that molecular forces produce. For example, if a body is brought into contact with a hot gas, then the energy of the gas is transferred through collisions of gas molecules with body molecules.

The amount of heat is not a function of state, since Q depends on the path of the system's transition from one state to another. If the state of the system is given, but the transition process is not specified, then nothing can be said about the amount of heat that is received by the system. In this sense, one cannot speak of the amount of heat stored in the body.

Sometimes they talk about a body that has a reserve of thermal energy, this does not mean the amount of heat, but the internal energy of the body. Such a body is called a heat reservoir. Such "blunders" in terminology remained in science from the theory of caloric, however, like the term itself, the amount of heat. The theory of caloric considered heat as a kind of imponderable fluid that is contained in bodies and cannot be created or destroyed. There was a version of caloric conservation. From this point of view, it was logical to talk about the stock of heat in the body without regard to the process. Now in calorimetry one often argues as if the law of conservation of the quantity of heat were valid. So, for example, they act in the mathematical theory of heat conduction.

Due to the fact that heat is not a function of state, the designation $\delta Q$ is used for an infinitesimal amount of heat, and not $dQ$. This emphasizes that $\delta Q$ is not considered as a total differential, i.e. cannot always be represented as infinitesimal increments of state functions (only in special cases, for example, in isochoric and isobaric processes). It is generally accepted that heat is positive if the system receives it, and negative otherwise.

What is heat capacity

Let us now consider what heat capacity is.

Definition

The amount of heat transferred to the body in order to heat it by 1K is the heat capacity of the body (system). Usually denoted by "C":

\[C=\frac(\delta Q)(dT)\left(1\right).\]

Heat capacity per unit body mass:

specific heat. m -- body weight.

Heat capacity per unit molar mass of a body:

molar heat capacity. $\nu $ - amount of substance (number of moles of substance), $\mu $ - molar mass of substance.

The average heat capacity $\left\langle C\right\rangle $ in the temperature range from $T_1$ to $T_2\ $ is:

\[\left\langle C\right\rangle =\frac(Q)(T_2-T_1)\ \left(4\right).\]

The relationship between the average heat capacity of a body and its "simple" heat capacity is expressed as:

\[\left\langle C\right\rangle =\frac(1)(T_2-T_1)\int\limits^(T_2)_(T_1)(CdT)\ \left(5\right).\]

We see that the heat capacity is defined through the concept of "heat".

As already noted, the amount of heat supplied to the system depends on the process. Accordingly, it turns out that the heat capacity also depends on the process. Therefore, the formula for determining the heat capacity (1) should be refined and written as:

\[С_V=(\left(\frac(\delta Q)(dT)\right))_V,\ С_p=(\left(\frac(\delta Q)(dT)\right))_p(6)\ ]

heat capacity (gas) in constant volume and at constant pressure.

Thus, the heat capacity in the general case characterizes both the properties of the body and the conditions under which the body is heated. If the heating conditions are determined, then the heat capacity becomes a characteristic of the properties of the body. We see such heat capacities in reference tables. Heat capacities in processes at constant pressure and constant volume are state functions.

Example 1

Task: An ideal gas whose molecule has the number of degrees of freedom equal to i was expanded according to the law: $p=aV,$where $a=const.$ Find the molar heat capacity in this process.

\[\delta Q=dU+\delta A=\frac(i)(2)\nu RdT+pdV\left(1.2\right).\]

Since the gas is ideal, we use the Mendeleev-Claperon equation and the process equation to convert elementary work and obtain an expression for it in terms of temperature:

So, the element of work looks like:

\[\delta A=pdV=aVdV=\frac(\nu RdT)(2)\left(1.4\right).\]

Substitute (1.4) into (1.2), we get:

\[\delta Q=\nu c_(\mu )dT=\frac(i)(2)\nu RdT+\frac(\nu RdT)(2)\left(1.5\right).\]

We express the molar heat capacity:

Answer: The molar heat capacity in a given process has the form: $c_(\mu )=\frac(R)(2)\left(i+1\right).$

Example 2

Task: Find the change in the amount of heat of an ideal gas in the process p$V^n=const$ (such a process is called polytropic), if the number of degrees of freedom of the gas molecule is equal to i, the change in temperature in the process $\triangle T$, the amount of substance $\nu $ .

The basis for solving the problem will be the expression:

\[\triangle Q=C\triangle T\ \left(2.1\right).\]

Hence, it is necessary to find C (heat capacity in a given process). We use the first law of thermodynamics:

\[\delta Q=dU+pdV=\frac(i)(2)\nu RdT+pdV=CdT\to C=\frac(i)(2)\nu R+\frac(pdV)(dT)\ \ left(2.2\right).\]

Find $\frac(dV)(dT)$ using the process equation and the Mendeleev-Claperon equation:

Let us substitute the pressure and volume from (2.3.) into the equation of the given process, we obtain the polytropic equation in the parameters $V,T$:

In this case:

\[\frac(dV)(dT)=B"\cdot \frac(1)(1-n)T^(\frac(n)(1-n))\left(2.5\right).\] \ \ \[\triangle Q=C\triangle T=\nu R\left(\frac(i)(2)+\frac(1)(1-n)\right)\triangle T\left(2.8\right) .\]

Answer: The change in the amount of heat of an ideal gas in the process is given by the formula: $\triangle Q=\nu R\left(\frac(i)(2)+\frac(1)(1-n)\right)\triangle T$.

It is known that the supply of heat to the working fluid in any process is accompanied by a change in temperature. The ratio of the heat supplied (removed) in a given process to a change in temperature is called heat capacity of the body.

where dQ is the elementary amount of heat

dT - elementary temperature change.

The heat capacity is numerically equal to the amount of heat that must be supplied to the system in order to raise the temperature by 1 degree under given conditions. Measured in [J/K].

The amount of heat supplied to the working fluid is always proportional to the amount of the working fluid. For example, the amount of heat required to heat a brick and a brick wall by 1 degree is not the same, therefore, for comparison, specific heat capacities are introduced, attributing the supplied heat to a unit of the working fluid. Depending on the quantitative unit of the body to which heat is supplied in thermodynamics, mass, volume and molar heat capacities are distinguished.

Mass heat capacity is the heat capacity per unit mass of the working fluid,

The amount of heat required to heat 1 kg of gas by 1 K is called mass heat capacity.

The unit of mass heat capacity is J/(kg K). Mass heat capacity is also called specific heat capacity.

Volumetric heat capacity- heat capacity per unit volume of the working fluid,

The amount of heat required to heat 1 m 3 of gas by 1 K is called volumetric heat capacity.

Volumetric heat capacity is measured in J / (m 3 K).

Molar heat capacity- heat capacity, related to the amount of the working fluid,

where n is the amount of gas in moles.

The amount of heat required to heat 1 mole of gas by 1 K is called the molar heat capacity.

Molar heat capacity is measured in J / (mol × K).

Mass and molar heat capacities are related by the following relation:

or C m \u003d mc, where m is the molar mass

The heat capacity depends on the process conditions. Therefore, the index is usually indicated in the expression for heat capacity X, which characterizes the type of heat transfer process.

Index X means that the process of supply (or removal) of heat goes on at a constant value of some parameter, for example, pressure, volume.

Among such processes, two are of greatest interest: one at a constant volume of gas, the other at a constant pressure. In accordance with this, heat capacities at constant volume C v and heat capacity at constant pressure C p are distinguished.

1) The heat capacity at a constant volume is equal to the ratio of the amount of heat dQ to the temperature change dT of the body in an isochoric process (V = const):

;

2) The heat capacity at constant pressure is equal to the ratio of the amount of heat dQ to the temperature change dT of the body in an isobaric process (Р = const):

To understand the essence of these processes, consider an example.

Let there be two cylinders containing 1 kg of the same gas at the same temperature. One cylinder is completely closed (V = const), the other cylinder is closed from above by a piston, which exerts a constant pressure P on the gas (P = const).

Let us bring to each cylinder such an amount of heat Q that the temperature of the gas in them rises from T 1 to T 2 by 1K. In the first cylinder, the gas did not do the work of expansion, i.e. the amount of heat supplied will be

Q v \u003d c v (T 2 - T 1),

here the index v - means that the heat is supplied to the gas in a process with a constant volume.

In the second cylinder, in addition to the temperature increase by 1K, there was also a movement of the loaded piston (the gas changed volume), i.e. expansion work has been done. The amount of heat supplied in this case is determined from the expression:

Q p \u003d c p (T 2 - T 1)

Here, the index p - means that heat is supplied to the gas in a process with constant pressure.

The total amount of heat Q p will be greater than Q v by an amount corresponding to the work of overcoming external forces:

where R is the work of expansion of 1 kg of gas with an increase in temperature by 1K at T 2 - T 1 \u003d 1K.

Hence С р - С v = R

If we place in the cylinder not 1 kg of gas, but 1 mol, then the expression will take the form

Сm Р - Сm v = R m , where

R m - universal gas constant.

This expression is called Mayer's equations.

Along with the difference C p - C v in thermodynamic studies and practical calculations, the ratio of heat capacities C p and C v, which is called the adiabatic index, is widely used.

k \u003d C p / C v.

In molecular - kinetic theory, to determine k, the following formula is given k \u003d 1 + 2 / n,

where n is the number of degrees of freedom of molecular motion (for monatomic gases n = 3, for diatomic gases n = 5, for three or more atomic gases n = 6).

The change in internal energy by doing work is characterized by the amount of work, i.e. work is a measure of the change in internal energy in a given process. The change in the internal energy of a body during heat transfer is characterized by a quantity called the amount of heat.

is the change in the internal energy of the body in the process of heat transfer without doing work. The amount of heat is denoted by the letter Q .

Work, internal energy and the amount of heat are measured in the same units - joules ( J), like any other form of energy.

In thermal measurements, a special unit of energy, the calorie ( feces), equal to the amount of heat required to raise the temperature of 1 gram of water by 1 degree Celsius (more precisely, from 19.5 to 20.5 ° C). This unit, in particular, is currently used in calculating the consumption of heat (thermal energy) in apartment buildings. Empirically, the mechanical equivalent of heat has been established - the ratio between calories and joules: 1 cal = 4.2 J.

When a body transfers a certain amount of heat without doing work, its internal energy increases, if a body gives off a certain amount of heat, then its internal energy decreases.

If you pour 100 g of water into two identical vessels, and 400 g into another at the same temperature and put them on the same burners, then the water in the first vessel will boil earlier. Thus, the greater the mass of the body, the greater the amount of heat it needs to heat up. The same goes for cooling.

The amount of heat required to heat a body also depends on the kind of substance from which this body is made. This dependence of the amount of heat required to heat the body on the type of substance is characterized by a physical quantity called specific heat capacity substances.

- this is a physical quantity equal to the amount of heat that must be reported to 1 kg of a substance to heat it by 1 ° C (or 1 K). The same amount of heat is given off by 1 kg of a substance when cooled by 1 °C.

The specific heat capacity is denoted by the letter with. The unit of specific heat capacity is 1 J/kg °C or 1 J/kg °K.

The values of the specific heat capacity of substances are determined experimentally. Liquids have a higher specific heat capacity than metals; Water has the highest specific heat capacity, gold has a very small specific heat capacity.

Since the amount of heat is equal to the change in the internal energy of the body, we can say that the specific heat capacity shows how much the internal energy changes 1 kg substance when its temperature changes 1 °C. In particular, the internal energy of 1 kg of lead, when it is heated by 1 °C, increases by 140 J, and when it is cooled, it decreases by 140 J.

Q required to heat the body mass m temperature t 1 °С up to temperature t 2 °С, is equal to the product of the specific heat capacity of the substance, body mass and the difference between the final and initial temperatures, i.e.

Q \u003d c ∙ m (t 2 - t 1)

According to the same formula, the amount of heat that the body gives off when cooled is also calculated. Only in this case should the final temperature be subtracted from the initial temperature, i.e. Subtract the smaller temperature from the larger temperature.