Question

At what temperature, pressure remaining unchanged, will the rms velocity of a gas be half its value at \(0^{\circ} \mathrm{C}\) ? (A) \(204.75 \mathrm{~K}\) (B) \(204.75^{\circ} \mathrm{C}\) (C) \(-204.75 \mathrm{~K}\) (D) \(-204.75^{\circ} \mathrm{C}\)

Short answer

The temperature at which the root mean square (rms) velocity of a gas is half its value at \(0^{\circ} \mathrm{C}\) is approximately \(-204.75^{\circ}\mathrm{C}\) (option D).

Step-by-step solution

Write down the formula for rms velocity.

The formula for the root mean square velocity is given by: \(v_{rms} = \sqrt{\frac{3kT}{m}}\)

Calculate the ratio of the rms velocities.

We are given that the rms velocity at the unknown temperature \(T_2\) should be half of its value at \(T_1 = 273.15 \mathrm{~K}\). To find the ratio of the rms velocities, divide the rms velocity at \(T_2\) by the rms velocity at \(T_1\): \(\frac{v_{rms}(T_2)}{v_{rms}(T_1)} = \frac{1}{2}\) Plug in the formulas for the rms velocity for both temperatures: \(\frac{\sqrt{\frac{3kT_2}{m}}}{\sqrt{\frac{3kT_1}{m}}} = \frac{1}{2}\) Since the constant and the mass of the gas molecules are the same, we can cancel out the terms: \(\frac{\sqrt{T_2}}{\sqrt{T_1}} = \frac{1}{2}\)

Solve for the unknown temperature.

To solve for \(T_2\), we'll square both sides of the equation: \(T_2 = \frac{T_1}{4}\) Now, substitute the value of \(T_1 = 273.15 \mathrm{~K}\) and solve for \(T_2\): \(T_2 = \frac{273.15 \mathrm{~K}}{4} = 68.2875 \mathrm{~K}\) Now we need to convert the answer to Celsius: \(T_2^{\circ} \mathrm{C} = 68.2875 \mathrm{~K} - 273.15 \mathrm{~K} = -204.8625^{\circ}\mathrm{C}\) The final answer is approximately \(-204.75^{\circ}\mathrm{C}\), which corresponds to option (D).

Key concepts

Temperature Dependence

Temperature plays a pivotal role in the properties of gases, particularly influencing the speed or velocity of their molecules. The root mean square (rms) velocity, which is a type of average velocity of gas molecules, is directly related to temperature. The formula is:\[v_{rms} = \sqrt{\frac{3kT}{m}} \]Here, \(v_{rms}\) is the rms velocity, \(k\) is the Boltzmann constant, \(T\) is the temperature in Kelvin, and \(m\) is the mass of the gas molecule. This equation shows that:

• The rms velocity is proportional to the square root of the temperature.
• A higher temperature increases the rms velocity, causing the gas molecules to move faster.

When temperature changes, it alters the energy of the molecules. Hence, when we want to change the rms velocity of a gas and keep pressure constant, adjusting the temperature is key. Understanding this relationship is crucial for mastering gas behavior.

Ideal Gas Law

The Ideal Gas Law is a fundamental principle that describes the behavior of ideal gases under various conditions. It connects four essential properties: pressure \(P\), volume \(V\), temperature \(T\), and the number of moles \(n\) of the gas. The law is expressed by the equation:\[ PV = nRT \]Where \(R\) is the universal gas constant. In this law:

• When temperature is altered, either pressure or volume must change to maintain the relationship, unless otherwise constrained.
• If temperature increases and volume is constant, then the pressure must increase.

The law allows scientists and engineers to predict how a gas will respond to changes in its environment. By maintaining constant pressure, as in the given problem, a change in temperature directly influences other properties like velocity.

Kinetic Theory of Gases

The kinetic theory of gases provides a microscopic understanding of how gases behave, emphasizing the motion of their molecules. It describes gases as a large number of very small particles (molecules) that are in constant random motion. The theory helps to explain:

• The pressure of a gas as a result of collisions between molecules and the walls of a container.
• The temperature of a gas being related to the average kinetic energy of its molecules.

According to this theory, when the temperature of a gas increases, the kinetic energy of its molecules also increases, causing them to move faster. This is why gas pressure increases if the volume is constant, as per the Ideal Gas Law. Understanding kinetic theory offers insight into why temperature affects properties like rms velocity, as increased molecular speed equates to higher energy levels, thus illustrating the interconnectedness of temperature, velocity, and kinetic energy in gases.