"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 ___________."

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Accepted Answer

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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 \times (30)^2}{3 \times 1 \times 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.

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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 ___________.

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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 ___________.

Solution

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