$70 cal$ of heat is required to raise the temperature of $2$ moles of an ideal gas at constant pressure from $30^{\circ} C$ to $35^{\circ} C$ . The amount of heat required to raise the temperature of the same gas through same range of temperature ( $30^{\circ} C$ to $35^{\circ} C$ ) at constant volume is
Given $(γ = 7 / 5)$

30 cal

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50 cal

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70 cal

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90 cal

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Solution

The correct option is B $50 cal$
Heat absorbed at constant pressure to increase the temperature by $Δ T$ ,
$Q_{p} = n C_{p} Δ T . . . (i)$
$C_{p} =$ molar heat capacity at constant pressure
$\Rightarrow$ Heat required to raise the temperature of same gas by $Δ T$ at constant volume is,
$Q_{V} = n C_{V} Δ T . . . (i i)$
$C_{V} =$ molar heat capacity at constant volume
$\Rightarrow \frac{Q_{V}}{Q_{p}} = \frac{C_{V}}{C_{p}}$
$Q_{V} = Q_{p} \times \frac{C_{V}}{C_{p}}$
$Q_{V} = 70 \times \frac{1}{γ}$
$∴ Q_{V} = 70 \times \frac{5}{7} = 50 cal$

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70 calories of heat are required to raise the temperature of 2 moles of an ideal gas at constant pressure from $30^{\circ} C t o 35^{\circ} C$ . The amount of heat required to raise the temperature of same gas through the same range ( $30^{\circ} C t o 35^{\circ} C$ ) at constant volume (R = 2 cal/mol/K)