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Chapter 8: Problem 12

If you have any two gases in different containers that are the same size at the same pressure and same temperature, what is true about the moles of each gas? Why is this true?

Short Answer

Expert verified
Under the given conditions where both gases have the same pressure, temperature, and equal volume containers, the moles of both gases are equal (i.e., they have the same number of moles), according to the Ideal Gas Law. This is true due to the direct proportionality of pressure, volume, and temperature to the moles of an ideal gas.

Step by step solution

01

Ideal Gas Law Equation

The Ideal Gas Law is a mathematical equation relating the volume, temperature, pressure, and amount of substance (in moles) for an ideal gas. The Ideal Gas Law equation is as follows: \(PV = nRT\) Where: - P is the pressure of the gas, - V is the volume of the gas, - n is the number of moles of the gas, - R is the gas constant (8.314 J/(mol K)), - T is the temperature of the gas in Kelvin.
02

Apply the given conditions

In this exercise, we know that both gases are at the same pressure, same temperature, and in containers of the same size (equal volume). Let's denote the conditions on gas 1 as \(P_1\), \(V_1\), \(n_1\), \(T_1\) and the conditions on gas 2 as \(P_2\), \(V_2\), \(n_2\), \(T_2\). Given that \(P_1 = P_2\), \(V_1 = V_2\), and \(T_1 = T_2\), let's apply the Ideal Gas Law to both gases: Gas 1: \(P_1V_1 = n_1RT_1\) Gas 2: \(P_2V_2 = n_2RT_2\)
03

Comparing the moles of both gases

Since the pressure, volume, and temperature of both gases are equal, we can now compare the moles of both gases. Divide the equation for gas 1 by the equation for gas 2: \(\frac{P_1V_1}{P_2V_2} = \frac{n_1RT_1}{n_2RT_2}\) Since \(P_1 = P_2\), \(V_1 = V_2\), and \(T_1 = T_2\), this simplifies to: \(\frac{n_1}{n_2} = 1\) Which means: \(n_1 = n_2\)
04

Conclusion

Under the given conditions where both gases have the same pressure, temperature, and equal volume containers, the moles of both gases are equal (i.e., they have the same number of moles), according to the Ideal Gas Law. This is true due to the direct proportionality of pressure, volume, and temperature to the moles of an ideal gas.

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Most popular questions from this chapter

A bicycle tire is filled with air to a pressure of \(75\) psi at a temperature of \(19^{\circ} \mathrm{C}\). Riding the bike on asphalt on a hot day increases the temperature of the tire to \(58^{\circ} \mathrm{C}\). The volume of the tire increases by \(4.0 \% .\) What is the new pressure in the bicycle tire?

Draw a qualitative graph to show how the first property varies with the second in each of the following (assume 1 mole of an ideal gas and \(T\) in kelvin). a. \(P V\) versus \(V\) with constant \(T\) b. \(P\) versus \(T\) with constant \(V\) c. \(T\) versus \(V\) with constant \(P\) d. \(P\) versus \(V\) with constant \(T\) e. \(P\) versus \(1 / V\) with constant \(T\) f. \(P V / T\) versus \(P\)

The average lung capacity of a human is \(6.0 \mathrm{~L}\). How many moles of air are in your lungs when you are in the following situations? a. At sea level \((T=298 \mathrm{~K}, P=1.00 \mathrm{~atm})\). b. \(10 . \mathrm{m}\) below water \((T=298 \mathrm{~K}, P=1.97 \mathrm{~atm})\). c. At the top of Mount Everest \((T=200 . \mathrm{K}, P=0.296 \mathrm{~atm})\).

You have a sealed, flexible balloon filled with argon gas. The atmospheric pressure is \(1.00\) 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. b. Above what temperature would you heat the balloon so that it would float?

An organic compound contains \(\mathrm{C}, \mathrm{H}, \mathrm{N},\) and \(\mathrm{O} .\) Combustion of \(0.1023 \mathrm{g}\) of the compound in excess oxygen yielded \(0.2766 \mathrm{g} \mathrm{CO}_{2}\) and \(0.0991 \mathrm{g} \mathrm{H}_{2} \mathrm{O} .\) A sample of \(0.4831 \mathrm{g}\) of the compound was analyzed for nitrogen by the Dumas method (see Exercise 129 ). At \(\mathrm{STP}, 27.6 \mathrm{mL}\) of dry \(\mathrm{N}_{2}\) was obtained. In a third experiment, the density of the compound as a gas was found to be \(4.02 \mathrm{g} / \mathrm{L}\) at \(127^{\circ} \mathrm{C}\) and \(256\) torr. What are the empirical and molecular formulas of the compound?
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