Question 796: Thermodynamics & KTG

Question

The r.m.s. velocity of hydrogen at S.T.P. is ' $u$ ' $\mathrm{m} / \mathrm{s}$. If the gas is heated at constant pressure till its volume becomes three times, then the final temperature of the gas and the r.m.s. speed are respectively

Accepted Answer

To determine the final temperature and the root mean square (r.m.s.) velocity of the hydrogen gas when its volume becomes three times at constant pressure, the following relationships and concepts are used:
Ideal Gas Law at Constant Pressure:
At constant pressure, the volume ($V$) of an ideal gas varies directly with its absolute temperature ($T$):
$$ \frac{V_1}{T_1} = \frac{V_2}{T_2} $$
Given that the volume triples ($V_2 = 3V_1$), and initially, the gas is at Standard Temperature and Pressure (STP) where $T_1 = 273 	ext{ K}$, we can solve for the final temperature $T_2$:
$$ \frac{V_1}{273} = \frac{3V_1}{T_2} $$
Simplifying gives:
$$ T_2 = 819 \, 	ext{K} $$
RMS Velocity Calculation:
The r.m.s. velocity of a gas is given by:
$$ u = \sqrt{\frac{3kT}{m}} $$
where $k$ is the Boltzmann constant and $m$ is the molecular mass. The r.m.s. velocity is proportional to the square root of the temperature ($u \propto \sqrt{T}$).
Initially, the velocity $u_1 = u$ at $T_1 = 273 	ext{ K}$. For the final velocity $u_2$ at $T_2 = 819 	ext{ K}$:
$$ u_2 = u \sqrt{\frac{T_2}{T_1}} = u \sqrt{\frac{819}{273}} = u \sqrt{3} $$
Thus, the final temperature of the gas is $819 \, 	ext{K}$ and the final r.m.s. speed is $u \sqrt{3} \, 	ext{m/s}$.
The correct option is
Option A: $819 \, 	ext{K}, \, u \sqrt{3} \, 	ext{m/s}$
.

Answer

$819 \mathrm{~K},(\sqrt{3}) \mathrm{um} / \mathrm{s}$

Answer

$1092 \mathrm{~K}, 3 \mathrm{um} / \mathrm{s}$

Answer

$819 \mathrm{~K}, \frac{\mathrm{u}}{\sqrt{3}} \mathrm{~m} / \mathrm{s}$

Answer

$1092 \mathrm{~K}, \frac{\mathrm{u}}{3} \mathrm{~m} / \mathrm{s}$

Thermodynamics & KTG

The r.m.s. velocity of hydrogen at S.T.P. is ' u ' \mathrm{m} / \mathrm{s}. If the gas is heated at constant pressure till its volume becomes three times, then the final temperature of the gas and the r.m.s. speed are respectively

Choose the correct answer: