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Chapter 19: Problem 70

A gas expands at constant pressure from \(3.00 \mathrm{~L}\) at \(15.0^{\circ} \mathrm{C}\) until the volume is \(4.00 \mathrm{~L}\). What is the final temperature of the gas?

Short Answer

Expert verified
Answer: The final temperature of the gas is \(384.2 \mathrm{~K}\).

Step by step solution

01

Convert temperatures to Kelvin

First, we need to convert the given initial temperature from Celsius to Kelvin using the formula: K = 掳C + 273.15 Initial temperature = \(15 ^\circ \mathrm{C}\) \(T_1 = 15 + 273.15 = 288.15 \mathrm{~K}\)
02

Set up Charles' Law formula

Using the Charles' Law formula for gas under constant pressure: \(\frac{V_1}{T_1} = \frac{V_2}{T_2}\) Given: \(V_1 = 3.00 \mathrm{~L}\), \(V_2 = 4.00 \mathrm{~L}\), and \(T_1 = 288.15 \mathrm{~K}\) Our goal is to find the final temperature, \(T_2\).
03

Substitute the values and solve for the final temperature

Now we'll substitute the given values into the formula and solve for the final temperature: \(\frac{3.00 \mathrm{~L}}{288.15 \mathrm{~K}} = \frac{4.00 \mathrm{~L}}{T_2}\) To solve for \(T_2\), we'll cross-multiply and divide: \(T_2 = \frac{4.00 \mathrm{~L} \cdot 288.15 \mathrm{~K}}{3.00 \mathrm{~L}}\)
04

Calculate the final temperature

Now, we'll perform the calculations: \(T_2 = \frac{4.00 \cdot 288.15}{3.00}\) \(T_2 = 384.2 \mathrm{~K}\) The final temperature of the gas is \(384.2 \mathrm{~K}\).

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Most popular questions from this chapter

A tire on a car is inflated to a gauge pressure of \(32 \mathrm{lb} / \mathrm{in}^{2}\) at a temperature of \(27^{\circ} \mathrm{C}\). After the car is driven for \(30 \mathrm{mi}\), the pressure has increased to \(34 \mathrm{lb} / \mathrm{in}^{2} .\) What is the temperature of the air inside the tire at this point? a) \(40^{\circ} \mathrm{C}\) b) \(23^{\circ} \mathrm{C}\) c) \(32^{\circ} \mathrm{C}\) d) \(54^{\circ} \mathrm{C}\)

An ideal gas may expand from an initial pressure, \(p_{\mathrm{i}},\) and volume, \(V_{\mathrm{i}},\) to a final volume, \(V_{\mathrm{f}}\), isothermally, adiabatically, or isobarically. For which type of process is the heat that is added to the gas the largest? (Assume that \(p_{i}, V_{i}\) and \(V_{f}\) are the same for each process.) a) isothermal process b) adiabatic process c) isobaric process d) All the processes have the same heat flow.

The compression and rarefaction associated with a sound wave propogating in a gas are so much faster than the flow of heat in the gas that they can be treated as adiabatic processes. a) Find the speed of sound, \(v_{s}\), in an ideal gas of molar mass \(M\). b) In accord with Einstein's refinement of Newtonian mechanics, \(v_{\mathrm{s}}\) cannot exceed the speed of light in vacuum, \(c\). This fact implies a maximum temperature for an ideal gas. Find this temperature. c) Evaluate the maximum temperature of part (b) for monatomic hydrogen gas (H). d) What happens to the hydrogen at this maximum temperature?

As noted in the text, the speed distribution of molecules in the Earth's atmosphere has a significant impact on its composition. a) What is the average speed of a nitrogen molecule in the atmosphere, at a temperature of \(18.0^{\circ} \mathrm{C}\) and a (partial) pressure of \(78.8 \mathrm{kPa} ?\) b) What is the average speed of a hydrogen molecule at the same temperature and pressure?

Show that the adiabatic bulk modulus, defined as \(B=-V(d P / d V),\) for an ideal gas is equal to \(\gamma P\).
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