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Chapter 5: Problem 42

A balloon is filled to a volume of \(7.00 \times 10^{2} \mathrm{~mL}\) at a temperature of \(20.0^{\circ} \mathrm{C}\). The balloon is then cooled at constant pressure to a temperature of \(1.00 \times 10^{2} \mathrm{~K}\). What is the final volume of the balloon?

Short Answer

Expert verified
The final volume of the balloon after cooling is approximately \(238.8 \mathrm{~mL}\).

Step by step solution

01

Write down the Charles' Law formula.

Charles' Law states that at constant pressure, the volume of a gas is directly proportional to its temperature in Kelvin. The formula for Charles' Law is: \(V_1/T_1 = V_2/T_2\) where, - \(V_1\) is the initial volume of the gas - \(T_1\) is the initial temperature in Kelvin - \(V_2\) is the final volume of the gas - \(T_2\) is the final temperature in Kelvin
02

Convert the given temperatures to Kelvin.

The initial temperature is given in Celsius, and we need to convert it to Kelvin using the formula: \(T(K) = T(^\circ C) + 273.15\) Initial temperature in Kelvin = \(20.0 + 273.15 = 293.15\ \mathrm{K}\) The final temperature is already given in Kelvin.
03

Substitute the given values into the Charles' Law formula.

Now, we have all the necessary values, and we can substitute them into the Charles' Law formula: \(\frac{7.00 \times 10^{2} \mathrm{~mL}}{293.15 \mathrm{~K}} = \frac{V_2}{1.00 \times 10^{2} \mathrm{~K}}\)
04

Solve for the final volume \(V_2\).

To find the final volume, we can cross-multiply and solve for \(V_2\): \(V_2 = \frac{7.00 \times 10^{2} \mathrm{~mL} \times 1.00 \times 10^{2} \mathrm{~K}}{293.15 \mathrm{~K}}\)
05

Calculate the final volume.

Now, compute the final volume: \(V_2 = \frac{7.00 \times 10^{2} \mathrm{~mL} \times 1.00 \times 10^{2} \mathrm{~K}}{293.15 \mathrm{~K}} = 238.8 \mathrm{~mL}\) The final volume of the balloon after cooling is approximately \(238.8 \mathrm{~mL}\).

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Most popular questions from this chapter

Nitrous oxide \(\left(\mathrm{N}_{2} \mathrm{O}\right)\) is a gas that is commonly used to help sedate patients in medicine and dentistry due to its mild anesthetic and analgesic properties, as well as the fact that it is nonflammable. If a cylinder of \(\mathrm{N}_{2} \mathrm{O}\) is at \(32.4 \mathrm{~atm}\) and has a volume of 5.0 \(\mathrm{L}\) at \(298 \mathrm{~K}\), how many moles of \(\mathrm{N}_{2} \mathrm{O}\) gas are in the cylinder? What volume would the gas take up if the entire contents of the cylinder were allowed to escape into a larger container that keeps the pressure constant at \(1.00 \mathrm{~atm}\) ? Assume the temperature remains at \(298 \mathrm{~K}\).

Sulfur trioxide, \(\mathrm{SO}_{3}\), is produced in enormous quantities each year for use in the synthesis of sulfuric acid. $$ \begin{aligned} \mathrm{S}(s)+\mathrm{O}_{2}(g) & \longrightarrow \mathrm{SO}_{2}(g) \\ 2 \mathrm{SO}_{2}(g)+\mathrm{O}_{2}(g) & \longrightarrow 2 \mathrm{SO}_{3}(g) \end{aligned} $$ What volume of \(\mathrm{O}_{2}(g)\) at \(350 .{ }^{\circ} \mathrm{C}\) and a pressure of \(5.25 \mathrm{~atm}\) is needed to completely convert \(5.00 \mathrm{~g}\) sulfur to sulfur trioxide?

You have a sealed, flexible balloon filled with argon gas. The atmospheric pressure is \(1.00 \mathrm{~atm}\) and the temperature is \(25^{\circ} \mathrm{C}\). Assume that air has a mole fraction of nitrogen of \(0.790\), the rest being oxygen. a. Explain why the balloon would float when heated. Make sure to discuss which factors change and which remain constant, and why this matters. Be complete. b. Above what temperature would you heat the balloon so that it would float?

Which of the following statements is(are) true? For the false statements, correct them. a. At constant temperature, the lighter the gas molecules, the faster the average velocity of the gas molecules. b. At constant temperature, the heavier the gas molecules, the larger the average kinetic energy of the gas molecules. c. A real gas behaves most ideally when the container volume is relatively large and the gas molecules are moving relatively quickly. d. As temperature increases, the effect of interparticle interactions on gas behavior is increased. e. At constant \(V\) and \(T\), as gas molecules are added into a container, the number of collisions per unit area increases resulting in a higher pressure. f. The kinetic molecular theory predicts that pressure is inversely proportional to temperature at constant volume and moles of gas.

Air bags are activated when a severe impact causes a steel ball to compress a spring and electrically ignite a detonator cap. This causes sodium azide \(\left(\mathrm{NaN}_{3}\right)\) to decompose explosively according to the following reaction: $$ 2 \mathrm{NaN}_{3}(s) \longrightarrow 2 \mathrm{Na}(s)+3 \mathrm{~N}_{2}(g) $$ What mass of \(\operatorname{NaN}_{3}(s)\) must be reacted to inflate an air bag to \(70.0 \mathrm{~L}\) at STP?
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