Question

H calories of heat are required to raise the temperature of 1 mole of a monatomic gas from $200^\circ C$ to $300^\circ C$ at constant volume. The amount of heat required (in calories) to raise the temperature of 2 moles of diatomic gas from $200^\circ C$ to $250^\circ C$ at constant pressure is:
A) $\dfrac{4}{3}H$.
B) $\dfrac{5}{3}H$.
C) $2H$.
D) $\dfrac{7}{3}H$.

Answer 1

Hint: The heat is a form of energy which can increase the temperature of liquid, solid or gas. The heat gained at constant volume is when volume doesn’t change and heat gained at constant pressure is when the pressure is constant and doesn’t change.

Formula used:The formula of the heat at constant volume is given by,
$ \Rightarrow {Q_v} = n{C_v}\Delta T$
Where heat gained is${Q_v}$, the number of molecules of the substance is n, the specific heat at constant volume is ${C_v}$ and the change in temperature is$\Delta T$.
The formula of the heat at constant pressure is given by,
$ \Rightarrow {Q_p} = n{C_p}\Delta T$
Where heat gained is${Q_p}$, the number of mole of the substance is n, the specific heat at constant pressure is ${C_p}$ and the change in temperature is$\Delta T$.

Complete step by step solution:
It is given in the problem that H calories of heat are required to raise the temperature of 1 mole of a monatomic gas from $200^\circ C$ to $300^\circ C$ at constant volume then we need to find the heat required (in calories) to raise the temperature of 2 moles of diatomic gas from $200^\circ C$ to $250^\circ C$ at constant pressure.
The formula of the heat at constant volume is given by,
$ \Rightarrow {Q_v} = n{C_v}\Delta T$
Where heat gained is${Q_v}$, the number of molecules of the substance is n, the specific heat at constant volume is ${C_v}$ and the change in temperature is$\Delta T$.
As the heat is H and the change of the temperature is from $200^\circ C$ to $300^\circ C$ and the number of moles is 1.
$ \Rightarrow {Q_v} = n{C_v}\Delta T$
The value of specific heat at constant volume is${C_v} = \dfrac{{3R}}{2}$.
$ \Rightarrow {Q_v} = 1 \times \dfrac{{3R}}{2} \times \left( {300 - 200} \right)$
$ \Rightarrow H = 150R$………eq. (1)
Now heat required for a diatomic gas having change of temperature from $200^\circ C$ to $250^\circ C$ at constant pressure and the specific heat at constant pressure is equal to${C_p} = \dfrac{{7R}}{2}$.
The formula of the heat at constant pressure is given by,
$ \Rightarrow {Q_p} = n{C_p}\Delta T$
Where heat gained is${Q_p}$, the number of mole of the substance is n, the specific heat at constant pressure is ${C_p}$ and the change in temperature is$\Delta T$.
$ \Rightarrow {Q_p} = 2 \times \dfrac{{7R}}{2} \times \left( {250 - 200} \right)$
$ \Rightarrow {H_1} = 350R$………eq. (2)
Comparing the equation (1) and equation (2) we get,
$ \Rightarrow {H_1} = \dfrac{7}{3}H$.
The amount of heat required (in calories) to raise the temperature of 2 moles of diatomic gas from $200^\circ C$ to $250^\circ C$ at constant pressure is$\dfrac{7}{3}H$.

The correct answer for this problem is option D.

Note: It is advisable for students to understand and remember the formula of the heat gained at constant volume and constant pressure as it is very useful in solving the problems like these. The SI unit of heat is Joules but it can be also expressed in calories.