Question

The quantity of heat required to heat one mole of a monoatomic gas through one degree K at constant pressure is:
A 3.5R
B 2.5R
C 1.5R
D none of these

Answer 1

Hint: The above problem can be solved by using the principles of the thermodynamics. The heat required to heat the gas depends on the number of moles of gas, specific heat at constant pressure and difference in the temperature. The specific heat at constant pressure is calculated by using the Mayer’s formula, which states that the difference between the specific heat at constant pressure and specific heat at constant volume is equal to the universal gas constant.

Complete step by step answer
Given: The temperature difference of the monatomic gas is $\Delta T = 1\;{\text{K}}$, the number of moles of the monatomic gas is $n = 1$.
The specific heat at constant volume of the monatomic gas is given as:
$\Rightarrow {C_v} = \dfrac{{fR}}{2}......\left( 1 \right)$
Here, f is the degree of freedom of monoatomic gas and its value is 3.
Substitute 3 for f in the expression (1) to find the specific heat at constant volume.
$\Rightarrow {C_v} = \dfrac{{\left( 3 \right)R}}{2}$
$\Rightarrow {C_v} = 1.5R$
The specific heat at constant pressure of the monatomic gas is given as:
$\Rightarrow {C_p} - {C_v} = R......\left( 2 \right)$
Substitute $1.5R$ for ${C_v}$ in the expression (2) to find the specific heat at constant pressure.
$\Rightarrow {C_p} - 1.5R = R$
$\Rightarrow {C_p} = R + 1.5R$
$\Rightarrow {C_p} = 2.5R$
The expression to calculate the heat required to heat the one mole of the gas at constant pressure is:
$\Rightarrow Q = n{C_P}\Delta T......\left( 3 \right)$
Substitute 1 for n, 2.5R for ${C_p}$ and $1\;{\text{K}}$ for $\Delta T$ in the expression (3) to find the heat required to heat the one mole of the gas at constant pressure.
$\Rightarrow Q = \left( {1\;{\text{mole}}} \right)\left( {2.5R} \right)\left( {1\;{\text{K}}} \right)$
$\Rightarrow Q = 2.5R$

Thus, the heat required to heat the one mole of the gas at constant pressure is $2.5R$ and the option (B) is correct answer.

Note: The specific heat at constant pressure is defined as the heat that raises the temperature of the gas by 1 K at constant pressure. The degree of freedom of the diatomic gas is 5 and the degree of freedom of the polyatomic gas is 6. The ratio of the specific heat at constant pressure to specific heat at constant volume is called the specific heat ratio.