In thermodynamics, when studying transitions from the initial to the final state of a certain system, it is important to know the thermal effect of the process. The concept of heat capacity is closely related to this effect. In this article, we consider the question of what is meant by the isochoric heat capacity of a gas.
Perfect gas
An ideal gas is gas whose particles are considered material points, that is, they do not have dimensions, but have mass, and in which all internal energy consists exclusively of the kinetic energy of the movement of molecules and atoms.
Ideally, any real gas will never satisfy the described model, since its particles still have some linear dimensions and interact with each other using weak van der Waals bonds or chemical bonds of a different type. However, at low pressures and high temperatures, the distances between the molecules are large, and their kinetic energy exceeds the potential tens of times. All this makes it possible to apply with a high degree of accuracy the ideal model for real gases.
Gas internal energy
The internal energy of any system is a physical characteristic that is equal to the sum of the potential and kinetic energy. Since the potential energy can be neglected in ideal gases, we can write for them the equality:
U = E k .
Where E k is the energy of the kinetic system. Using the molecular-kinetic theory and applying the universal Clapeyron-Mendeleev equation of state, it is easy to obtain an expression for U. It is written below:
U = z / 2 * n * R * T.
Here T, R and n are absolute temperature, gas constant and amount of substance, respectively. The value of z is an integer indicating the number of degrees of freedom that a gas molecule has.
Isobaric and isochoric heat capacity
In physics, the heat capacity is the amount of heat that must be provided to the system under study in order to heat it by one kelvin. The reverse definition is also true, that is, heat capacity is the amount of heat that the system emits when it is cooled by one kelvin.
The easiest way for a system to determine isochoric heat capacity. Under it is understood the heat capacity at a constant volume. Since the system does not perform work under such conditions, all the energy is spent on increasing internal energy reserves. Denote the isochoric heat capacity by the symbol C V , then we can write:
dU = C V * dT.
That is, a change in the internal energy of a system is directly proportional to a change in its temperature. If we compare this expression with the equality written in the previous paragraph, then we arrive at the formula for C V in an ideal gas:
With V = z / 2 * n * R.
In practice, this value is inconvenient to use, since it depends on the amount of substance in the system. Therefore, the concept of specific isochoric heat capacity was introduced, that is, a value that is calculated either per 1 mole of gas or per 1 kg. Denote the first value by the symbol C V n , the second by the symbol C V m . For them, you can write the following formulas:
C V n = z / 2 * R;
C V m = z / 2 * R / M.
Here M is the molar mass.
Isobaric is the heat capacity while maintaining a constant pressure in the system. An example of such a process is the expansion of gas in a cylinder under a piston when it is heated. Unlike isochoric, during the isobaric process, the heat supplied to the system is consumed to increase internal energy and to perform mechanical work, that is:
H = dU + P * dV.
The enthalpy of the isobaric process is the product of the isobaric heat capacity and the temperature change in the system, that is:
H = C P * dT.
If we consider expansion at a constant pressure of 1 mol of gas, then the first law of thermodynamics will be written in the form:
C P n * dT = C V n * dT + R * dT.
The last term is obtained from the Clapeyron-Mendeleev equation. From this equality follows the relationship between isobaric and isochoric heat capacities:
C P n = C V n + R.
For an ideal gas, the specific molar heat capacity at constant pressure is always greater than the corresponding isochoric characteristic by R = 8.314 J / (mol * K).
Degrees of freedom of molecules and heat capacity
Let us write out again the formula for the specific molar isochoric heat capacity:
C V n = z / 2 * R.
In the case of a monatomic gas, the quantity z = 3, since atoms in space can only move in three independent directions.
If we are talking about a gas consisting of diatomic molecules, for example, oxygen O 2 or hydrogen H 2 , then, in addition to translational motion, these molecules can also rotate around two mutually perpendicular axes, that is, z will be equal to 5.
In the case of more complex molecules, z = 6 should be used to determine C V n .