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

JEE Main · 2020 · 03 Sep 2020 · Shift 2 · Heat & Thermodynamics

To raise the temperature of a certain mass of gas by

5 0^{\circ} C

at a constant pressure, 160 calories of heat is required. When the same mass of gas is cooled by

10 0^{\circ} 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 be ideal)?

(A)7
(B)5
(C)3
(D)6✓ Correct

Answer: (D)

Solution

Heating at constant pressure,

n C_{p} Δ T = 160 cal

with

Δ T = 50

gives

n C_{p} = 3.2

. Cooling at constant volume,

n C_{v} Δ T = 240 cal

with

Δ T = 100

gives

n C_{v} = 2.4

. Their ratio is

γ = \frac{C _{p}}{C _{v}} = \frac{3.2}{2.4} = \frac{4}{3}

. Since

γ = 1 + \frac{2}{f}

for

f

degrees of freedom,

\frac{4}{3} = 1 + \frac{2}{f}

gives

\frac{2}{f} = \frac{1}{3}

, so

f = 6

JEE Main · 2020 · 03 Sep 2020 · Shift 2 · Single correct option

Practise more: all JEE Main PYQs on Heat & Thermodynamics →

More JEE Main questions on Heat & Thermodynamics

A cylindrical tube $A B$ of length $l$ , closed at both ends contains an ideal gas of 1 mol having molecular weight $M$ . The tube is rotated in a horizontal plane with constant angular velocity $ω$ about an axis perpendicular to $A B$ and passing through the edge at end A, as s …
JEE Main 2026View solution →
Rods $x$ and $y$ of equal dimensions but of different materials are joined as shown in figure. Temperatures of end points $A$ and $F$ are maintained at $10 0^{\circ} C$ and $4 0^{\circ} C$ respectively. Given the thermal conductivity of rod x is three …
JEE Main 2026View solution →
The volume of an ideal gas increases 8 times and temperature becomes $(1/4)^{th}$ of initial temperature during a reversible change. If there is no exchange of heat in this process ( $Δ Q = 0$ ) then identify the gas from the following options (Assuming …
JEE Main 2026View solution →
Consider two boxes containing ideal gases A and B such that their temperatures, pressures and number densities are same. The molecular size of A is half of that of B and mass of molecule A is four times that of B. If the collision frequency in gas B is …
JEE Main 2026View solution →
An insulated cylinder of volume $60 cm^{3}$ is filled with a gas at $2 7^{\circ} C$ and 2 atmospheric pressure. Then the gas is compressed making the final volume as $20 cm^{3}$ while allowing the temperature to rise to $7 7^{\circ} C$ . The fin …
JEE Main 2026View solution →
The r.m.s speed of oxygen molecules at $4 7^{\circ} C$ is equal to that of the hydrogen molecules kept at $____$ $^{\circ} C$ . (Mass of oxygen molecule/mass of hydrogen molecule=32/2)
JEE Main 2026View solution →