26. (JEE Main 2019 (Online) 10th January Evening Slot )

Two kg of a monoatomic gas is at a pressure of 4 $\times$ 10⁴ N/m². The density of the gas is 8 kg/m³. What is the order of energy of the gas due to its thermal motion ?

(A) 10⁴ J

(B) 10³ J

(C) 10⁵ J

(D) 10⁶ J

Correct answer is (A)

Thermal energy of N molecule

= N $(\frac{3}{2} k T)$

= $\frac{N}{N_{A}} \frac{3}{2}$ RT

= $\frac{3}{2}$ (nRT)

= $\frac{3}{2}$ PV

= $\frac{3}{2}$ P $(\frac{m}{8})$

= $\frac{3}{2}$ $\times$ 4 $\times$ 10⁴ $\times$ $\frac{2}{8}$

= 1.5 $\times$ 10⁴

order will 10⁴

27. (JEE Main 2017 (Online) 9th April Morning Slot )

N moles of a diatomic gas in a cylinder are at a temperature T. Heat is supplied to the cylinder such that the temperature remains constant but n moles of the diatomic gas get converted into monoatomic gas. What is the change in the total kinetic energy of the gas ?

(A) $\frac{1}{2}$ nRT

(B) 0

(C) $\frac{3}{2}$ nRT

(D) $\frac{5}{2}$ nRT

Correct answer is (A)

Initial kinetic energy of N mole of diatomic gas,

K_i = N $\frac{5}{2}$ RT

Kinetic energy of n mole of monoatomic gas
= n $\frac{3}{2}$ RT

When n mole of diatomic gas converted into monoatomic gas then remaining diatomic gas = (N $-$ n)

Find kinetic energy,

K_F = (2m) $\frac{3}{2}$ RT + (N $-$ n) $\frac{5}{2}$ RT

= $\frac{1}{2}$ nRT + $\frac{5}{2}$ NRT

$∴$ Change in kinetic energy,

$Δ$ K = K_f $-$ K_i = $\frac{1}{2}$ nRT

28. (AIEEE 2011 )

A thermally insulated vessel contains an ideal gas of molecular mass $M$ and ratio of specific heats $γ .$ It is moving with speed $v$ and it's suddenly brought to rest. Assuming no heat is lost to the surroundings, Its temperature increases by:

(A) $\frac{(γ - 1)}{2 γ R} M v^{2} K$

(B) $\frac{γ M^{2} v}{2 R} K$

(C) $\frac{(γ - 1)}{2 R} M v^{2} K$

(D) $\frac{(γ - 1)}{2 (γ + 1) R} M v^{2} K$

Correct answer is (C)

Here, work done is zero.
So, loss in kinetic energy $=$ change in internal energy of gas
$\frac{1}{2} m v^{2} = n C_{v} Δ T = n \frac{R}{γ - 1} Δ T$
$\frac{1}{2} m v^{2} = \frac{m}{M} \frac{R}{γ - 1} Δ T$
$∴$ $Δ T = \frac{M v^{2} (γ - 1)}{2 R} K$

29. (AIEEE 2011 )

Three perfect gases at absolute temperatures $T_{1}, T_{2}$ and $T_{3}$ are mixed. The masses of molecules are $m_{1}, m_{2}$ and $m_{3}$ and the number of molecules are $n_{1},$ $n_{2}$ and $n_{3}$ respectively. Assuming no loss of energy, the final temperature of the mixture is:

(A) $\frac{n_{1} T_{1} + n_{2} T_{2} + n_{3} T_{3}}{n_{1} + n_{2} + n_{3}}$

(B) $\frac{n_{1} T_{1}^{2} + n_{2} T_{2}^{2} + n_{3} T_{3}^{2}}{n_{1} T_{1} + n_{2} T_{2} + n_{3} T_{3}}$

(C) $\frac{n_{1}^{2} T_{1}^{2} + n_{2}^{2} T_{2}^{2} + n_{3}^{2} T_{3}^{2}}{n_{1} T_{1} + n_{2} T_{2} + n_{3} T_{3}}$

(D) $\frac{(T_{1} + T_{2} + T_{3})}{3}$

Correct answer is (A)

Number of moles of first gas $= \frac{n_{1}}{N_{A}}$
Number of moles of second gas $= \frac{n_{2}}{N_{A}}$
Number of moles of third gas $= \frac{n_{3}}{N_{A}}$
If there is no loss of energy then
$P_{1} V_{1} + P_{2} V_{2} + P_{3} V_{3} = P V$
$\frac{n_{1}}{N_{A}} R T_{1} + \frac{n_{2}}{N_{A}} R T_{2} + \frac{n_{3}}{N_{A}} R T_{3}$
$= \frac{n_{1} + n_{2} + n_{3}}{N_{A}} R T_{m i x}$
$\Rightarrow T_{m i x} = \frac{n_{1} T_{1} + n_{2} T_{2} + n_{3} T_{3}}{n_{1} + n_{2} + n_{3}}$

30. (AIEEE 2009 )

One $k g$ of a diatomic gas is at a pressure of $8 \times 10^{4} N / m^{2} .$ The density of the gas is $4 k g / m^{3}$ . What is the energy of the gas due to its thermal motion ?

(A) $5 \times 10^{4} J$

(B) $6 \times 10^{4} J$

(C) $7 \times 10^{4} J$

(D) $3 \times 10^{4} J$

Correct answer is (A)

$V o l u m e = \frac{m a s s}{d e n s i t y} = \frac{1}{4} m^{3}$
$K . E = \frac{5}{2} P V$
$= \frac{5}{2} \times 8 \times 10^{4} \times \frac{1}{4}$
$= 5 \times 10^{4} J$

Kinetic Theory of Gases