Question 559: Thermodynamics & KTG

Question

A monoatomic gas of mass 4.0 u is kept in an insulated container. Container is moving with velocity 30 m/s. If container is suddenly stopped then change in temperature of the gas (R = gas constant) is $${x \over {3R}}$$. Value of x is ___________.

Accepted Answer

To find the change in temperature of a monoatomic gas when the container it is in is suddenly stopped, we can follow these steps:
Relationship Between Kinetic Energy and Internal Energy Change:
The kinetic energy lost by the container when it stops is converted into the internal energy of the gas:
$    \Delta KE = \Delta U    $
Expression for Internal Energy Change:
The change in internal energy can be expressed in terms of the change in temperature:
$    \Delta U = n C_v \Delta T    $
where $ n $ is the number of moles, $ C_v $ is the molar heat capacity at constant volume, and $ \Delta T $ is the change in temperature.
Using the Kinetic Energy Formula for the Change:
For a monoatomic gas, the molar heat capacity at constant volume $ C_v $ is $\frac{3R}{2}$. Thus, we equate the kinetic energy to the change in internal energy:
$    \frac{1}{2} m v^2 = \frac{3}{2} n R \Delta T    $
Solving for $\Delta T$:
Rearrange the equation to solve for the change in temperature:
$    \Delta T = \frac{m v^2}{3 n R}    $
Substituting Given Values:
Here, mass $ m = 4 $ u, velocity $ v = 30 $ m/s, and the number of moles $ n = 1 $. Substitute these into the equation:
$    \Delta T = \frac{4 	imes (30)^2}{3 	imes 1 	imes R}    $
Simplifying:
$    \Delta T = \frac{3600}{R}    $
Relating to Given Expression:
According to the problem, $\Delta T = \frac{x}{3R}$. Therefore, equating:
$    \frac{x}{3R} = \frac{3600}{R}    $
Solve for $x$:
Simplifying gives:
$    x = 3600    $
Thus, the value of $x$ is 3600.

Thermodynamics & KTG

A monoatomic gas of mass 4.0 u is kept in an insulated container. Container is moving with velocity 30 m/s. If container is suddenly stopped then change in temperature of the gas (R = gas constant) is ${x \over {3R}}$. Value of x is ___________.

Integer / Numeric Answer Type