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Chapter 13: Problem 40

A flight attendant wants to change the temperature of the air in the cabin from \(18^{\circ} \mathrm{C}\) to \(24^{\circ} \mathrm{C}\) without changing the number of moles of air per \(\mathrm{m}^{3} .\) What fractional change in pressure would be required?

Short Answer

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Question: A flight attendant wants to change the temperature of the air in the cabin without changing the number of moles of air per cubic meter. Calculate the fractional change in pressure required to change the temperature from 18掳C to 24掳C. Answer: The fractional change in pressure required is approximately 0.0206, or about a 2.06% increase in pressure.

Step by step solution

01

Write down the Ideal Gas Law formula

The Ideal Gas Law formula is given by \(PV=nRT\), where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature in Kelvin.
02

Convert temperatures to Kelvin

Given temperatures in Celsius, we must convert them to Kelvin by adding 273.15. So, the initial temperature \(T_{1} = 18 + 273.15 = 291.15 \;\mathrm{K}\), and the final temperature \(T_{2} = 24 + 273.15 = 297.15 \;\mathrm{K}\).
03

Find the ratio of initial and final pressures

Since the number of moles of air per \(\mathrm{m}^{3}\) doesn't change, the relationship between initial and final pressures is given by the ratio of initial and final temperatures, assuming constant volume (V) and number of moles (n). This can be expressed using the Ideal Gas Law formula as follows: \(\frac{P_{1}}{P_{2}} = \frac{nR T_{1}}{nR T_{2}}\) Since the volume, number of moles, and gas constant are the same for both temperatures, we can simplify: \(\frac{P_{1}}{P_{2}} = \frac{T_{1}}{T_{2}}\)
04

Calculate the fractional change in pressure

Now, we can calculate the fractional change in pressure using the ratio found in step 3: Fractional change in pressure = \(\frac{P_{2} - P_{1}}{P_{1}} = \frac{T_{2}}{T_{1}} - 1 = \frac{297.15}{291.15} - 1\) Fractional change in pressure \(\approx 0.0206\) Therefore, the fractional change in pressure required is approximately 0.0206, or approximately a 2.06% increase in pressure.

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