First slide

Specific heat and degrees of freedom

Question

To raise the temperature of a certain mass of gas by $50^{0}$ C at a constant pressure, 160 calories of heat is required. When the same mass of gas is cooled by $100^{0}$ C at constant volume, 240 calories of heat is released. How many degrees of freedom does each molecule of this gas have (assume gas to ideal)?

Moderate

A

3
B

6
C

5
D

7

Solution

At constant pressure $d Q = m c_{p} d T$
$160 = m c_{P} (50)$
At constant volume
$\begin{array}{l} dQ = m c_{V} dT \\ 240 = m c_{V} (100) \\ \Rightarrow \Rightarrow \frac{c_{P} (50)}{c_{v (100)}} = \frac{160}{240} \\ \frac{γ}{2} = \frac{2}{3} (∵ γ = \frac{c_{P}}{c_{v}} = \frac{C_{P}}{C_{v}}) \\ γ = \frac{4}{3} \\ f = \frac{2}{γ - 1} = \frac{2}{\frac{4}{3} - 1} = 6 \end{array}$

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