Question

The specific heat of air at constant volume is $0.172\text{Cal}{{\text{g}}^{\text{-1}}}{{\text{C}}^{\text{-1}}}$. The change in internal energy when 5g of air is heated from 0C to 4C at constant volume is:
A. 28.8 J
B. 14.4 J
C. 7.2 J
D. 3.51 J

Answer 1

Hint: When some heat is given to a gas at a constant volume, only its internal energy changes. The change in the internal energy at constant volume of the gas is given as $\Delta U=m{{C}_{p}}\Delta T$. Use this formula to find the change in internal energy.

Formula used:
$\Delta U=m{{C}_{p}}\Delta T$

Complete step by step answer:
The change in internal energy of a gas at constant volume is given as $\Delta U=m{{C}_{p}}\Delta T$.
Here, m is the mass of the gas, ${{C}_{p}}$ is the specific heat of the gas at constant volume and $\Delta T$ is the change in the temperature of the gas.
In this case, m = 5g, ${{C}_{p}}=0.172\text{Cal}{{\text{g}}^{\text{-1}}}{{\text{C}}^{\text{-1}}}$ and $\Delta T$= 4C.
Hence, $\Delta U=5\times 0.172\times 4=3.44Cal$
We know that 1 cal = 4.2 J.
$\Rightarrow \Delta U=3.44\times 4.2J=14.448J\approx 14.4J$
Therefore, the change in the internal energy of air is 14.4 J.
Hence, the correct option is B.
Additional Information: According to the second law of thermodynamics, when some heat is given to a gas, the gas consumes this energy and raises some amount of its internal energy and does some work.
i.e $Q=W+\Delta U$.
Here, Q is the amount of heat given to or taken from the system. W is the work done by the system (gas). $\Delta U$ is the change in internal energy of the system.
 Work done by a gas is given as $W=\int{PdV}$,
where P is the pressure of the gas and dV is the change in the volume of the gas.
In the given case, air is at constant volume. This means that there is change in the volume of the gas. Therefore. dV = 0. This implies that W = 0.

Note: Note that the formula for the change in the internal energy that we used is applicable only for constant volume of the gas.
For general cases, change in internal energy is given as $\Delta U=\dfrac{f}{2}\mu R\Delta T$ .
Here, f is the degree of freedom of the gas, $\mu $ is the number of moles of the gas, R is the universal gas constant and $\Delta T$ is the change in temperature of the gas.