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Chapter 5: 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
The moles of each gas in the two containers are equal. This is true because, according to the Ideal Gas Law, the relationship between the pressure, volume, temperature, and number of moles must remain constant for two gases given the same conditions. Since both containers share the same pressure, volume, and temperature, the equation \(PV = nRT\) is satisfied, and thus the moles of each gas are equal.

Step by step solution

01

Recognize the given information

Given two containers that are the same size, both containers have the same volume (V). Additionally, both gases have the same pressure (P) and temperature (T).
02

Use the Ideal Gas Law

The Ideal Gas Law is given by the equation PV = nRT, where P is the pressure, V is the volume, n is the number of moles, T is the temperature, and R is the gas constant. By substituting the values given in the problem, we can find the moles of each gas.
03

Compare the two gases

Let the two containers be labeled A and B. Apply the Ideal Gas Law equation to both containers: Container A: \(P_A V_A = n_A R T_A\) Container B: \(P_B V_B = n_B R T_B\) Since both containers have the same pressure, volume, and temperature, we can write: \(P_A = P_B\) \(V_A = V_B\) \(T_A = T_B\)
04

Equate the Ideal Gas Law equations

We can set the two Ideal Gas Law equations equal to each other, since the pressure, volume, and temperature are the same for both containers: \(P_A V_A = n_A R T_A\) \(P_B V_B = n_B R T_B\) \(PV = n_A R T\) \(PV = n_B R T\)
05

Solve for the relationship between moles of each gas

Dividing the two equations, we find the relationship between the moles of each gas: \(\frac{n_A R T}{n_B R T} = \frac{PV}{PV}\) Canceling out the common terms, we are left with: \(\frac{n_A}{n_B} = \frac{PV}{PV}\) \(\frac{n_A}{n_B} = 1\) Thus, the moles of each gas (n_A and n_B) are equal. This is true because, according to the Ideal Gas Law, the relationship between the pressure, volume, temperature, and number of moles must remain constant for two gases given the same conditions.

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Most popular questions from this chapter

Which of the following statements is(are) true? a. If the number of moles of a gas is doubled, the volume will double, assuming the pressure and temperature of the gas remain constant b. If the temperature of a gas increases from \(25^{\circ} \mathrm{C}\) to \(50^{\circ} \mathrm{C},\) the volume of the gas would double, assuming that the pressure and the number of moles of gas remain constant. c. The device that measures atmospheric pressure is called a barometer. d. If the volume of a gas decreases by one half, then the pressure would double, assuming that the number of moles and the temperature of the gas remain constant.

As weather balloons rise from the earth鈥檚 surface, the pressure of the atmosphere becomes less, tending to cause the volume of the balloons to expand. However, the temperature is much lower in the upper atmosphere than at sea level. Would this temperature effect tend to make such a balloon expand or contract? Weather balloons do, in fact, expand as they rise. What does this tell you?

You have a helium balloon at 1.00 atm and \(25^{\circ} \mathrm{C} .\) You want to make a hot-air balloon with the same volume and same lift as the helium balloon. Assume air is 79.0\(\%\) nitrogen and 21.0\(\%\) oxygen by volume. The 鈥渓ift鈥� of a balloon is given by the difference between the mass of air displaced by the balloon and the mass of gas inside the balloon. a. Will the temperature in the hot-air balloon have to be higher or lower than \(25^{\circ} \mathrm{C} ?\) Explain. b. Calculate the temperature of the air required for the hot-air balloon to provide the same lift as the helium balloon at 1.00 atm and \(25^{\circ} \mathrm{C}\) . Assume atmospheric conditions are 1.00 atm and \(25^{\circ} \mathrm{C} .\)

If 27.1 g of \(\mathrm{Ar}(g)\) occupies a volume of 4.21 \(\mathrm{L}\) , what volume will 1.29 moles of \(\mathrm{Ne}(g)\) occupy at the same temperature and pressure?

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