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Chapter 9: Problem 1299

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

Expert verified
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

01

Write down the formula for rms velocity.

The formula for the root mean square velocity is given by: \(v_{rms} = \sqrt{\frac{3kT}{m}}\)
02

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: \(\frac{\sqrt{T_2}}{\sqrt{T_1}} = \frac{1}{2}\)
03

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).

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Most popular questions from this chapter

The temperature of an ideal gas is increased from \(27^{\circ} \mathrm{C}\) to \(927^{\circ} \mathrm{C}\). The root mean square speed of its molecules becomes (A) Four times (B) One-fourth (C) Half (D) Twice

The temperature of an ideal gas is increased from \(27^{\circ} \mathrm{C}\) to \(127^{\circ} \mathrm{C}\), then percentage increase in \(\mathrm{v}_{\mathrm{rms}}\) is (A) \(33 \%\) (B) \(11 \%\) (C) \(15.5 \%\) (D) \(37 \%\)

Hydrogen gas is filled in a balloon at \(20^{\circ} \mathrm{C}\). If temperature is made \(40^{\circ} \mathrm{C}\), pressure remaining the same what fraction of hydrogen will come out (A) \(0.75\) (B) \(0.07\) (C) \(0.25\) D) \(0.5\)

Air is pumped into an automobile tube upto a pressure of \(200 \mathrm{kPa}\) in the morning when the air temperature is \(22^{\circ} \mathrm{C}\). During the day, temperature rises to \(42^{\circ} \mathrm{C}\) and the tube expands by $2 \%$ The pressure of the air in the tube at this temperature will be approximately. (A) \(209 \mathrm{kPa}\) (B) \(206 \mathrm{kPa}\) (C) \(200 \mathrm{kPa}\) (D) \(212 \mathrm{kPa}\)

To what temperature should the hydrogen at room temperature $\left(27^{\circ} \mathrm{C}\right)$ be heated at constant pressure so that the rms velocity of its molecule becomes double of its previous value (A) \(927^{\circ} \mathrm{C}\) (B) \(600^{\circ} \mathrm{C}\) (C) \(108^{\circ} \mathrm{C}\) (D) \(1200^{\circ} \mathrm{C}\)
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