Question

The internal energy of air in a room of volume $50{m^3}$ at atmospheric pressure will be
A) $2.5 \times {10^7}\,erg$
B) $2.5 \times {10^7}\,joules$
C) $5.255 \times {10^7}\,joules$
D) $1.25 \times {10^7}\,joules$

Answer 1

Hint
The internal energy for the air is given by $U = \dfrac{f}{2}nRT$. The majority of gas is nitrogen and oxygen which are diatomic and have a degree of freedom of five. We will substitute the data in the expression and simplify to get the value of internal energy.

Complete step-by-step answer
The internal energy is about molecular freedom and termed as the kinetic internal energy. This is given by,
$U = \dfrac{f}{2}nRT$
Where, f is the degree of freedom, R is the universal gas constant and T is the temperature.
From the ideal gas equation,
$PV = nRT$
So, the energy equation can be modified as,
$U = \dfrac{f}{2}PV$
Air in a room contains mostly nitrogen and oxygen. Both are diatomic molecules so the degree of freedom is 5.
Given that,
$P = 1.013 \times {10^5}$
$V = 50{m^3}$
Substitute in the internal energy formula.
$U = \dfrac{5}{2} \times 1.013 \times {10^5} \times 50$
$U = 1.25 \times {10^7}J$.
Hence, internal energy is $1.25 \times {10^7}J$.
The correct option is (D).

Note
The term degree of freedom refers to the possible independent motions, a system can have. For monatomic gases, the degree of freedom is three. For diatomic gases, the degree of freedom is five. For triatomic gases, the degree of freedom is five for linear and six for non-linear molecules.