Question

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

Answer 1

Step 1: It is given to us that mass of the monoatomic gas $m=4u$, Velocity of the gas $v=30m/s$ .$R$ is gas constant and change in temperature after stopping movement is $∆T=\frac{x}{3R}$.

Step 2: We know that the kinetic energy of the container during movement is: $KE=\frac{1}{2}m{v}^2$ where, $m$ is mass and $v$ is velocity.

The internal energy of the container when not in movement is: $∆U=n{C}_v∆T$ where, $n$ is the number of atoms in the gas, $C_v$ is specific heat and$∆T$ is the change in temperature.

When the container stops, the kinetic energy will be equal to internal energy or $$\begin{gathered} ∆KE=∆U \\ \frac{1}{2}m{v}^2=n{C}_v∆T \end{gathered}$$

Step 3: The specific heat for monoatomic gas is $\frac{3}{2}R$for one unit change in temperature. Putting the given values in the above equation: $$\begin{gathered} \frac{1}{2}\times 4\times {\left(30\right)}^2=n\frac{3}{2}R∆T \\ 1800=n\frac{3}{2}R\frac{x}{3R}\left(∆T=\frac{x}{3R},n=1\right) \\ 1800\times 2=x \end{gathered}$$

Therefore, the value of $\mathit{x}\mathbf{=3600}$.