Question

Define ${C_p}$ and ${C_v}$. Derive the relation ${C_p} - {C_v} = R$

Answer 1

When heat is absorbed by a body, the temperature of the body increases. And when heat is lost, the temperature decreases. The temperature of an object is the measure of the entire Kinetic energy. of the particles that structure that object. So, when heat is absorbed by an object this heat gets translated into the Kinetic energy of the particles and as a result the temperature increases.

Complete step by step solution:
The definition of ${C_p}$ :
The ${C_p}$ is the amount of heat energy is released or absorbed by the unit mass of the substance with the constant pressure at change in temperature. In other words, in constant pressure the heat energy transfers between the system and its surroundings.

The definition of ${C_v}$ :
The ${C_v}$ is the amount of heat energy is released or absorbed by the unit mass of the substance with the constant volume at change in temperature. In other words, in constant volume the heat energy transfers between the system and its surroundings.

Relationship between ${C_p}$ and ${C_v}$ :
According to the first law of thermodynamics,
$q = nC\Delta T\,.................\left( 1 \right)$
Where $q$ is the heat, $n$ is the number of moles, $C$ molar heat capacity and $\Delta T$ is the change in temperature.
At constant pressure, in the equation (1), then
${q_p} = n{C_p}\Delta T$
The above equation is equal to the change in enthalpy, then
${q_p} = n{C_p}\Delta T = \Delta H\,..............\left( 2 \right)$
Similarly, at constant, volume, in equation (1), then
${q_v} = n{C_v}\Delta T$
The above equation is equal to the change in internal energy, then
${q_p} = n{C_p}\Delta T = \Delta U\,................\left( 3 \right)$
The formula for one mole of an ideal gas is,
$\Delta H = \Delta U + \Delta \left( {pv} \right)$ $\left( {pv = nRT} \right)$ (For one mole $n = 1$)
Then the above equation is written as,
$\Delta H = \Delta U + \Delta \left( {RT} \right)$
By rearranging the above equation, then
$\Delta H = \Delta U + R\Delta T\,................\left( 4 \right)$
By substituting the equation (2) and equation (3) in the equation (4), then
$n{C_p}\Delta T = n{C_v}\Delta T + R\Delta T$
Here, $n = 1$, then the above equation is written as,
${C_p}\Delta T = {C_v}\Delta T + R\Delta T$
By taking $\Delta T$ as a common term, then
${C_p} \times \Delta T = \left( {{C_v} + R} \right)\Delta T$
By cancelling the terms $\Delta T$ on both sides, then
${C_p} = {C_v} + R$
By rearranging the above equation, then
${C_p} - {C_v} = R$

Note: When the equation (2) and the equation (3) is substituted in the equation (4) and the mentioned the value for $n$ is one, because the equation is for one mole of the ideal gas, so the value for $n$ is one. The heat capacity is depending on the nature, size and composition of the system.