Question

The temperature of 5 mole of a gas which was held at constant volume was changed from 100°C to 120°C. The change in internal energy was found to be 80 J. The total heat capacity of the gas at constant volume will be equal to
A) 8 J/K
B) 0.8 J/K
C) 4.0 J/K
D) 0.4 J/K

Answer 1

We are provided with the number of moles of gas, change in temperature and internal energy. So, we can find the heat capacity using the formula for change in internal energy but the total heat capacity will be its product with the number of moles of the gas present.

Formula Used: $\Delta U = n{C_V}\Delta T$ where, $\Delta U$ is the change in internal energy, n is the number of moles, ${C_V}$ is the heat capacity at constant volume and $\Delta T$ is the change in temperature.

Complete step by step answer: Given:
Number of moles of gas (n) = 5
Change in temperature $\left( {\Delta T} \right)$ 🡪 from ${T_1} = 100$ to ${T_2} = 120$
$\Rightarrow \Delta T = {T_2} - {T_1} \\\ \Rightarrow \Delta T = 120 - 100 \\\ \Rightarrow \Delta T = {20^0}C \\\$

Change in internal energy $\left( {\Delta U} \right)$ = 80 J
Let the heat capacity at constant volume be ${C_V}$
Now, the internal energy can be given as:
$\Delta U = n{C_V}\Delta T$
Substituting the known value to find the value of ${C_V}$:
$80 = 5 \times {C_V} \times 20 \\\ \Rightarrow 80 = {C_V} \times 100 \\\ \Rightarrow {C_V} = \dfrac{{80}}{{100}} \\\ \Rightarrow {C_V} = 0.8 \\\$
The total heat capacity is given by its product with the number of moles. So:
${C_{total}} = n \times {C_V} \\\ \Rightarrow {C_{total}} = 5 \times 0.8 \\\ \Rightarrow {C_{total}} = 4.0 \\\$
The heat capacity is measured in the units: joule/kelvin (J/K)
Therefore, the total heat capacity of the gas at constant volume will be equal to 4.0 J/K and the correct option is C).

Note: We use delta $\left( \Delta \right)$ to denote the change in respective physical quantities.
Internal energy is the sum total of kinetic (due to their motion) and potential (due to their vibration) energies associated with the gas molecules and the heat capacity is the total internal energy of the system.
The process occurring here is called as a thermodynamic process and as it is at constant volume, it is known as isochoric thermodynamic process