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Chapter 13: Problem 72

a sealed cylinder contains a sample of ideal gas at a pressure of 2.0 atm. The rms speed of the molecules is \(v_{0} .\) If the rms speed is then reduced to \(0.90 v_{0},\) what is the pressure of the gas?

Short Answer

Expert verified
Answer: The new pressure of the gas is 1.62 atm.

Step by step solution

01

Identify the relevant equation

We know that rms speed is related to the pressure, volume, and temperature of the gas. The equation we will use is derived from the ideal gas law, \(PV = nRT\), and the rms speed is given by \(v_{rms} = \sqrt{\frac{3RT}{M}}\), where R is the gas constant, T is the temperature, and M is the molar mass of the gas. Since we are considering the same gas and same temperature, we can write the pressure as \(P = k \cdot v_{rms}^2\), where k is a constant, proportional to the gas's molar mass and temperature.
02

Calculate the constant k

First, we need to calculate the value of k. We can do this using the given pressure and rms speed: \(P_1 = 2.0 \, atm\) \(v_{rms1} = v_0\) Now, we can calculate k: \(2.0 = k\cdot v_0^2\) \(k = \frac{2.0}{v_0^2}\)
03

Find the new pressure with the reduced rms speed

Now, we need to find the new pressure (P2) when the rms speed of molecules is \(0.90 \cdot v_0\): \(v_{rms2} = 0.90 \cdot v_0\) Substitute this value into the equation for pressure, using the constant k we found in step 2: \(P_2 = k \cdot v_{rms2}^2\) \(P_2 = \frac{2.0}{v_0^2} \cdot (0.90 \cdot v_0)^2\) \(P_2 = \frac{2.0}{v_0^2} \cdot (0.81 \cdot v_0^2)\) \(P_2 = 0.81 \cdot 2.0 = 1.62 \, atm\)
04

Report the final answer

The pressure of gas when the rms speed is reduced to \(0.90 \cdot v_0\) is 1.62 atm.

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Most popular questions from this chapter

The volume of air taken in by a warm-blooded vertebrate in the Andes mountains is \(210 \mathrm{L} /\) day at standard temperature and pressure (i.e., \(\left.0^{\circ} \mathrm{C} \text { and } 1 \text { atm }\right) .\) If the air in the lungs is at \(39^{\circ} \mathrm{C},\) under a pressure of $450 \mathrm{mm} \mathrm{Hg}$ and we assume that the vertebrate takes in an average volume of \(100 \mathrm{cm}^{3}\) per breath at the temperature and pressure of its lungs, how many breaths does this vertebrate take per day?
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