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Chapter 21: Problem 39

A quantity of ideal gas at \(12.0^{\circ} \mathrm{C}\) and a pressure of \(108 \mathrm{kPa}\) occupies a volume of \(2.47 \mathrm{~m}^{3}\). (a) How many moles of the gas are present? \((b)\) If the pressure is now raised to \(316 \mathrm{kPa}\) and the temperature is raised to \(31.0^{\circ} \mathrm{C}\), how much volume will the gas now occupy? Assume there are no leaks.

Short Answer

Expert verified
There are 108 moles of gas present initially and the volume of gas after the changes is 0.879 m³.

Step by step solution

01

Convert temperatures to Kelvin and pressures to Pascal

First, convert the temperatures from degrees Celsius to Kelvin using the formula \(K = C + 273.15\) which gives \(T_{1} = 285.15 K\) and \(T_{2} = 304.15 K\). Then, convert the pressures from kPa to Pa (since 1 kPa = 1000 Pa) which gives \(P_{1} = 108000 Pa\) and \(P_{2} = 316000 Pa\).
02

Calculate the number of moles in the first scenario

Using the gas law equation \(PV = nRT\) and solve for \(n\), we get \(n_1 = \frac{P_1V_1}{RT_1}\). Substituting \(P_1 = 108000 Pa\), \(V_1 = 2.47 m^3\), \(R = 8.314 J/(mol*K)\), and \(T_1 = 285.15 K\), we calculate \(n_1 = 107.96\) moles.
03

Calculate the volume in the second scenario

Using the same gas law equation and solve for \(V\), we get \(V_2 = \frac{n_2RT_2}{P_2}\). Since there are no leaks, so \(n_2 = n_1\). Substituting \(n_2 = 107.96\) mol, \(R = 8.314 J/(mol*K)\), \(T_2 = 304.15 K\), and \(P_2 = 316000 Pa\), we calculate \(V_2 = 0.879\) m³.
04

Round off to the appropriate number of significant figures

Since the given values are to 3 significant figures, we should also maintain 3 significant figures in the answer. So, \(n = 108\) moles and \(V_2 = 0.879\) m³.

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Most popular questions from this chapter

Absolute zero is \(-273.15^{\circ} \mathrm{C}\). Find absolute zero on the Fahrenheit scale.

The best vacuum that can be attained in the laboratory corresponds to a pressure of about \(10^{-18} \mathrm{~atm}\), or \(1.01 \times 10^{-13} \mathrm{~Pa}\). How many molecules are there per cubic centimeter in such a vacuum at \(22^{\circ} \mathrm{C} ?\)

Repeat Exercise 1, except choose the new temperature scale \(\mathrm{Q}\) so that absolute zero is \(0^{\circ} \mathrm{Q}\) and \(T_{\mathrm{bp} \text { , water }}-T_{\text {mp, water }}=\) \(100 \mathrm{Q}^{\circ} .(a)\) What is the conversion formula from Celsius to Q? (b) What is \(T_{\text {bp, water }}\) and \(T_{\text {mp,water }}\) in Q? ( \(c\) ) This scale actually exists. What is the official name?

At what temperature is the Fahrenheit scale reading equal to (a) twice that of the Celsius and \((b)\) half that of the Celsius?

A rod is measured to be \(20.05 \mathrm{~cm}\) long using a steel ruler at a room temperature of \(20^{\circ} \mathrm{C}\). Both the rod and the ruler are placed in an oven at \(270^{\circ} \mathrm{C}\), where the rod now measures \(20.11 \mathrm{~cm}\) using the same rule. Calculate the coefficient of thermal expansion for the material of which the rod is made.
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