Question
The amount of heat energy required to raise the temperature of $$1g$$ of Helium at $$NTP,$$ from $${T_1}K$$ to $${T_2}K$$ is
A.
$$\frac{3}{2}{N_a}{k_B}\left( {{T_2} - {T_1}} \right)$$
B.
$$\frac{3}{4}{N_a}{k_B}\left( {{T_2} - {T_1}} \right)$$
C.
$$\frac{3}{4}{N_a}{k_B}\frac{{{T_2}}}{{{T_1}}}$$
D.
$$\frac{3}{8}{N_a}{k_B}\left( {{T_2} - {T_1}} \right)$$
Answer :
$$\frac{3}{8}{N_a}{k_B}\left( {{T_2} - {T_1}} \right)$$
Solution :
From first law of thermodynamics
$$\eqalign{
& \Delta Q = \Delta U + \Delta W = \frac{3}{2}.\frac{1}{4}R\left( {{T_2} - {T_1}} \right) + 0 \cr
& = \frac{3}{8}\;{N_a}{K_B}\left( {{T_2} - {T_1}} \right)\,\,\left[ {\because K = \frac{R}{N}} \right] \cr} $$