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Chapter 10: Problem 33

Gas is confined in a tank at a pressure of \(11.0 \mathrm{~atm}\) and a temperature of \(25.0^{\circ} \mathrm{C}\). If two thirds of the gas is withdrawn and the temperature is raised to \(75.0^{\circ} \mathrm{C}\), what is the new pressure in the tank?

Short Answer

Expert verified
After performing the calculations, \(P_2/3\) will give the new pressure in the tank after two thirds of the gas has been withdrawn and the temperature has been raised to \(75.0^{\circ} \mathrm{C}\).

Step by step solution

01

Convert temperatures to Kelvin

Convert the temperatures from Celsius to Kelvin. The Kelvin temperature is obtained by adding 273.15 to the Celsius temperature. So, \(T_1 = 25.0^{\circ} \mathrm{C} = 25.0 + 273.15 = 298.15 \mathrm{K}\) and \(T_2 = 75.0^{\circ} \mathrm{C} = 75.0 + 273.15 = 348.15 \mathrm{K}\)
02

Apply the gas law

We can apply the proportional relationship \(P_1/T_1 = P_2/T_2\) where \(P_1\) is the initial pressure, \(T_1\) is the initial temperature, \(P_2\) is the final pressure, and \(T_2\) is the final temperature. We are looking for \(P_2\) so we can rearrange the equation to be \(P_2 = P_1 * T_2 / T_1\). But we also know that we only have a third of the gas left, so the new pressure will be a third of this result.
03

Calculate the new pressure

Substitute \(P_1 = 11.0~atm\), \(T_1 = 298.15~K\), and \(T_2 = 348.15~K\) into the equation from step 2. This gives us \(P_2 = 11.0 * 348.15 / 298.15\). Since only a third of the gas remains, the new pressure is \(P_2/3\). Calculate the numerical value for \(P_2/3\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Thermodynamics
Understanding thermodynamics is crucial when dealing with gas behavior under varying conditions. Thermodynamics is a branch of physics concerned with heat and temperature and their relation to energy and work. It defines macroscopic variables, such as internal energy, entropy, and pressure, that partly describe a body of matter or radiation. It is also concerned with the laws of conservation of energy, which govern the flow of heat and work. In the context of gases, thermodynamics helps us comprehend how changes in temperature can impact gas molecules and their movements, which, in turn, affects the pressure within a given volume.
Pressure-Temperature Relationship
The pressure-temperature relationship, also known as Gay-Lussac's Law, states that the pressure of a gas is directly proportional to its temperature when the volume and the amount of gas are held constant. This means if you increase the temperature of a gas, its pressure will also increase if the gas is in a sealed container. Conversely, cooling the gas will decrease its pressure. This relationship is described by the formula \( P_1/T_1 = P_2/T_2 \), where \(P_1\) and \(P_2\) are the initial and final pressures, and \(T_1\) and \(T_2\) are the initial and final temperatures in Kelvin, respectively. Understanding this principle is key to solving problems involving temperature changes in gaseous systems, such as the textbook exercise we have focused on.
Ideal Gas Equation
The ideal gas equation is a fundamental equation that describes the state of an ideal gas. It combines several different gas laws and is expressed as \( PV = nRT \), where \(P\) stands for pressure, \(V\) is volume, \(n\) is the amount of gas in moles, \(R\) is the universal gas constant, and \(T\) is the temperature in Kelvin. This equation assumes that the gas particles are point particles that interact only through elastic collision. In practice, no gas is truly ideal, but many gases behave like ideal gases at a range of temperatures and pressures. The ideal gas law is invaluable for calculations involving gas reactions and thermodynamics, ensuring a strong foundation in understanding these concepts is vital for students tackling exercises such as the given scenario where the amount of gas changes while temperature and pressure conditions also change.

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Most popular questions from this chapter

A \(7.00-\mathrm{L}\) vessel contains \(3.50\) moles of ideal gas at a pressure of \(1.60 \times 10^{6} \mathrm{~Pa}\). Find (a) the temperature of the gas and (b) the average kinetic energy of a gas molecule in the vessel. (c) What additional information would you need if you were asked to find the average speed of a gas molecule?

A nurse measures the temperature of a patient to be \(43^{\circ} \mathrm{C}\). What is this temperature on the Fahrenheit scale? Do you think the patient is seriously ill? Explain.

The ideal gas law can be recast in terms of the density of a gas. (a) Use dimensional analysis to find an expression for the density \(\rho\) of a gas in terms of the number of moles \(n_{1}\) the volume \(V_{1}\) and the molecular weight \(M\) in kilograms per mole. (b) With the expression found in part (a), show that $$ P=\frac{\rho}{M} R T $$ for an ideal gas. (c) Find the density of the carbon dioxide atmosphere at the surface of Venus, where the pressure is \(90.0\) atm and the temperature is \(7.00 \times 10^{2} \mathrm{~K}\). (d) Would an evacuated steel shell of radius \(1.00 \mathrm{~m}\) and mass \(2.00 \times\) \(10^{2} \mathrm{~kg}\) rise or fall in such an atmosphere? Why?

Superman leaps in front of Lois Lane to save her from a volley of bullets. In a 1-minute interval, an automatic weapon fires 150 bullets, each of mass \(8.0 \mathrm{~g}\), at \(400 \mathrm{~m} / \mathrm{s}\). The bullets strike his mighty chest, which has an area of \(0.75 \mathrm{~m}^{2}\). Find the average force exerted on Superman's chest if the bullets bounce back after an elastic, head-on collision.

A \(20.0-\mathrm{L}\) tank of carbon dioxide gas \(\left(\overline{C O}_{2}\right)\) is at a pressure of \(9.50 \times 10^{5} \mathrm{~Pa}\) and temperature of \(19.0^{\circ} \mathrm{C}_{4}\) (a) Calculate the Lemperature of the gas in Kelvin. (b) Use the ideal gas law to calculate the number of moles of gas in the tank. (c) Use the periodic table to compute the molecular weight of carbon dioxide, expressing it in grams per mole. (d) Obtain the number of grams of carbon dioxide in the tank. (e) A fire breaks out, raising the ambicnt temperature by \(224,0 \mathrm{~K}\) while \(82.0 \mathrm{~g}\) of gas leak out of the tank. Calculate the new temperature and the number of moles of gas remaining in the tank. (f) Using a technique analogous to that in Example \(10.6 \mathrm{~b}\), find a symbolic expression for the final pressure, neglecting the change in volume of the tank. (g) Calculate the final pressure in the tank as a result of the fire and leakage.
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