Question

The specific heat of hydrogen gas at constant pressure is \[{{\text{C}}_p} = 3.4 \times {10^3}cal/k{g^0}C\] and at constant volume is \[{{\text{C}}_{\text{V}}} = 2.4 \times {10^3}cal/k{g^0}C\]. If one kilogram hydrogen gas is heated from \[{10^0}C\] to \[{20^0}C\] at constant pressure, the external work done on the gas to maintain is at constant pressure is
A. \[{10^5}{\text{cal}}\]
B. \[{10^4}{\text{cal}}\]
C. \[{10^3}{\text{cal}}\]
D. \[5 \times {10^3}{\text{cal}}\]

Answer 1

Hint:-To solve this problem, we have to apply the first law of thermodynamics. It states that the energy can neither be created nor destroyed but it can be converted from one form to another form. The first law of thermodynamics is basically the law of conservation of energy.
The mathematical form of first law of thermodynamics is, \[\Delta U = Q - W\]
Where, \[\Delta U\] is the change in internal energy of any system.
\[Q\] is the net heat transfer of the system.
W is the sum of work done by the system or on the system.

Complete step-by-step solution:
Given that:
The specific heat of hydrogen gas at constant pressure is, \[{{\text{C}}_p} = 3.4 \times {10^3}cal/k{g^0}C\]
The specific heat of hydrogen gas at constant volume is, \[{{\text{C}}_{\text{V}}} = 2.4 \times {10^3}cal/k{g^0}C\]
Mass of hydrogen gas is, \[{\text{m = 1kg}}\]
Heating temperature of hydrogen gas is, \[{10^0}C\] to \[{20^0}C\].
The heat transfer of the system at different conditions is calculated as, \[{Q_p} = m{C_p}\Delta t\]
The change in internal energy of the system is, \[{Q_v} = m{C_v}\Delta t\]
Therefore, the external work done on the gas at constant pressure is,
$ W = Q - \Delta U$
$ \Rightarrow m{C_p}\Delta t - m{C_v}\Delta t$
$ \Rightarrow m\left( {{C_p} - {C_v}} \right)\left( {{t_2} - {t_1}} \right)$
$ \Rightarrow 1\left( {3.4 \times {{10}^3} - 2.4 \times {{10}^3}} \right)\left( {20 - 10} \right)$
$ \Rightarrow {10^4}{\text{Cal}}$
Therefore, the external work done on the gas at constant pressure is, \[W = {10^4}{\text{cal}}\].
Hence, the correct option of this problem is, B.

Note:- Specific heat is a thermodynamic property which is related to the internal energy of the system. The specific heat at constant pressure and the specific heat at constant volume is an intensive property (does not depend on the mass) of the system.