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Chapter 10: Problem 3

Consider the sample of gas depicted here. What would the drawing look like if the volume and temperature remained constant while you removed enough of the gas to decrease the pressure by a factor of 2\(?[\) Section 10.3\(]\) \begin{equation}\begin{array}{l}{\text { (a) It would contain the same number of molecules. }} \\ {\text { (b) It would contain half as many molecules. }} \\\ {\text { (c) It would contain twice as many molecules. }} \\ {\text { (d) There is insufficient data to say. }}\end{array}\end{equation}

Short Answer

Expert verified
(b) It would contain half as many molecules.

Step by step solution

01

Apply Boyle's Law

As the volume V and the temperature T remain constant while the pressure P changes, we can apply Boyle's Law. Boyle's Law states that the product of the pressure and volume (P 脳 V) remains constant for an ideal gas when held at a constant temperature. Mathematically, it can be represented as: \(P_1V_1 = P_2V_2\) where \(P_1\) and \(P_2\) are the initial and final pressures, and \(V_1\) and \(V_2\) are the initial and final volumes.
02

Use the given condition to find the relationship between the initial and final pressures

Given that the pressure decreases by a factor of 2, we can relate the initial and final pressure as: \(P_2 = \dfrac{P_1}{2}\) Since the volume remains constant, we can say that \(V_1 = V_2\).
03

Determine the relationship between the initial and final number of molecules

Using the Ideal Gas Law, we can find the relationship between the initial and final number of molecules of the gas. The Ideal Gas Law is expressed as: \(PV = nRT\) 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. For the initial state: \(P_1V_1 = n_1RT_1\) For the final state: \(P_2V_2 = n_2RT_2\) As temperature and volume remain constant, we have: \(T_1 = T_2\) and \(V_1 = V_2\). Since \(P_1V_1 = P_2V_2\), we can write: \(n_1RT_1 = n_2RT_2\) Dividing both sides by RT: \(\dfrac{n_1T_1}{n_2T_2} = \dfrac{n_1}{n_2}\) As T is constant, we have: \(\dfrac{n_1}{n_2} = \dfrac{P_1V_1}{P_2V_2}\) Since the volume is constant, we can write the relation as: \(\dfrac{n_1}{n_2} = \dfrac{P_1}{P_2}\) Now substituting the value of \(P_2\) from step 2: \(\dfrac{n_1}{n_2} = \dfrac{P_1}{\frac{P_1}{2}}\) \(\dfrac{n_1}{n_2} = 2\) Here, \(n_2 = \dfrac{n_1}{2}\) which means there are now half as many molecules.
04

Choose the correct answer based on the analysis

Based on the relationship between the initial and final number of molecules, the correct answer is: (b) It would contain half as many molecules.

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Most popular questions from this chapter

A sample of \(5.00 \mathrm{~mL}\) of diethylether \(\left(\mathrm{C}_{2} \mathrm{H}_{3} \mathrm{OC}_{2} \mathrm{H}_{5}\right.\) ? density \(=0.7134 \mathrm{~g} / \mathrm{mL}\) ) is introduced into a \(6.00-\mathrm{L}\) vessel that already contains a mixture of \(\mathrm{N}_{2}\) and \(\mathrm{O}_{2}\), whose partial pressures are \(P_{\mathrm{N}_{2}}=0.751 \mathrm{~atm}\) and \(P_{\mathrm{O}_{1}}=0.208 \mathrm{~atm}\). The temperature is held at \(35.0^{\circ} \mathrm{C}\), and the diethylether totally evaporates. (a) Calculate the partial pressure of the diethylether. (b) Calculate the total pressure in the container.

Natural gas is very abundant in many Middle Eastern oil fields. However, the costs of shipping the gas to markets in other parts of the world are high because it is necessary to liquefy the gas, which is mainly methane and has a boiling point at atmospheric pressure of \(-164^{\circ} \mathrm{C}\). One possible strategy is to oxidize the methane to methanol, \(\mathrm{CH}_{3} \mathrm{OH}\), which has a boiling point of \(65^{\circ} \mathrm{C}\) and can therefore be shipped more readily. Suppose that \(10.7 \times 10^{9} \mathrm{ft}^{3}\) of methane at atmospheric pressure and \(25^{\circ} \mathrm{C}\) is oxidized to methanol. (a) What volume of methanol is formed if the density of \(\mathrm{CH}_{3} \mathrm{OH}\) is \(0.791 \mathrm{~g} / \mathrm{mL}\) ? (b) Write balanced chemical equations for the oxidations of methane and methanol to \(\mathrm{CO}_{2}(g)\) and \(\mathrm{H}_{2} \mathrm{O}(l)\). Calculate the total enthalpy change for complete combustion of the \(10.7 \times 10^{9} \mathrm{ft}^{3}\) of methane just described and for complete combustion of the equivalent amount of methanol, as calculated in part (a). (c) Methane, when liquefied, has a density of \(0.466 \mathrm{~g} / \mathrm{mL}\); the density of methanol at \(25^{\circ} \mathrm{C}\) is \(0.791 \mathrm{~g} / \mathrm{mL}\). Compare the enthalpy change upon combustion of a unit volume of liquid methane and liquid methanol. From the standpoint of energy production, which substance has the higher enthalpy of combustion per unit volume?

Cyclopropane, a gas used with oxygen as a general anesthetic, is composed of \(85.7 \% \mathrm{C}\) and \(14.3 \% \mathrm{H}\) by mass. (a) If \(1.56 \mathrm{~g}\) of cyclopropane has a volume of \(1.00 \mathrm{~L}\) at \(0.984 \mathrm{~atm}\) and \(50.0^{\circ} \mathrm{C}\), what is the molecular formula of cyclopropane? (b) Judging from its molecular formula, would you expect cyclopropane to deviate more or less than Ar from ideal-gas behavior at moderately high pressures and room temperature? Explain. (c) Would cyclopropane effuse through a pinhole faster or more slowly than methane, \(\mathrm{CH}_{4}\) ?

Consider the following gases, all at STP. Ne, \(\mathrm{SF}_{6}, \mathrm{~N}_{2}, \mathrm{CH}_{4}\). (a) Which gas is most likely to depart from the assumption of the kinetic- molecular theory that says there are no attractive or repulsive forces between molecules? (b) Which one is closest to an ideal gas in its behavior? (c) Which one has the highest root-mean-square molecular speed at a given temperature? (d) Which one has the highest total molecular volume relative to the space occupied by the gas? (e) Which has the highest average kinetic-molecular energy? (f) Which one would effuse more rapidly than \(\mathrm{N}_{2}\) ? (g) Which one would have the largest van der Waals \(b\) parameter?

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