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Chapter 22: Problem 8

A cylindrical container of length \(56.0 \mathrm{~cm}\) and diameter \(12.5 \mathrm{~cm}\) holds \(0.350\) moles of nitrogen gas at a pressure of \(2.05\) atm. Find the rms speed of the nitrogen molecules.

Short Answer

Expert verified
The root mean square speed of the nitrogen molecules is approximately 686 m/s.

Step by step solution

01

Gather information and convert units

Firstly, note down the given information. We have the number of moles of nitrogen gas \(n = 0.350\) moles, pressure \(P = 2.05\) atm and the gas constant \(R = 0.0821\) L atm K⁻¹mol⁻¹. The temperature will need to be calculated using the formula \(P = nRT/V\) where \(V\) is the volume of the gas calculated from the dimensions of the container given in the problem. The volume of a cylinder is given by \(V = \pi r²h\), where \(r\) is the radius, and \(h\) is the height. Substituting \(r = 6.25\) cm and \(h = 56.0\) cm (making sure to convert cm to L by multiplying with \(1.0 \times 10^{-3}\) L/cm), we find \(V \approx 0.686\) L.
02

Calculate the temperature

Rearrange the formula for \(P = nRT/V\) to find \(T = PV/nR\). Substituting \(P = 2.05\) atm, \(V = 0.686\) L, \(n = 0.350\) moles, and \(R = 0.0821\) L atm K⁻¹mol⁻¹, we find \(T \approx 151\) K.
03

Calculate the root mean square (rms) speed

The rms speed is given by \(v_{rms} = \sqrt{(3RT/M)}\), where \(M\) is the molar mass. The molar mass of nitrogen is \(M = 28.0\) g/mol = \(28.0 \times 10^{-3}\) kg/mol, and taking \(R = 8.314\) J K⁻¹ mol⁻¹ as the universal gas constant in these units, substituting \(T = 151\) K and \(M = 28.0 \times 10^{-3}\) kg/mol we find \(v_{rms} \approx 686\) m/s.

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Most popular questions from this chapter

The mass of the \(\mathrm{H}_{2}\) molecule is \(3.3 \times 10^{-24} \mathrm{~g}\). If \(1.6 \times 10^{23}\) hydrogen molecules per second strike \(2.0 \mathrm{~cm}^{2}\) of wall at an angle of \(55^{\circ}\) with the normal when moving with a speed of \(1.0 \times 10^{5} \mathrm{~cm} / \mathrm{s}\), what pressure do they exert on the wall?

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