Question

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

The moles of each gas are equal (n_1 = n_2) under the same pressure, volume, and temperature conditions. This is true because of the Ideal Gas Law, which dictates the relationship between the pressure, volume, temperature, and moles of a gas. In this case, since pressure, volume, and temperature are kept the same for both gases, their moles must also be equivalent.

Step-by-step solution

Recall the Ideal Gas Law

The Ideal Gas Law is given by the equation $PV=nRT$, where P is pressure, V is volume, n is the moles of gas, R is the ideal gas constant, and T is temperature.

Apply the given conditions to the Ideal Gas Law

Since the pressures, volumes, and temperatures of both gases are the same, we can rewrite the Ideal Gas Law for both gases as follows: For Gas 1: $P_1{V}_1=n_{1}R{T}_1$ For Gas 2: $P_2{V}_2=n_{2}R{T}_2$ Given that $P_1=P_2$, $V_1=V_2$, and $T_1=T_2$, we can now compare these two equations.

Compare the equations for both gases

Since the pressure, volume, and temperature are the same for both gases, the equations become: $PV=n_{1}R\cancel{T}$ $PV=n_{2}R\cancel{T}$ Since both equations are equal to the same value, we can equate them: $n_{1}RT=n_{2}RT$

Simplify the equation and find the relationship between moles

We can now divide both sides of the equation by RT to find the relationship between the moles of each gas: $n_1\cancel{R}\cancel{T}=n_2\cancel{R}\cancel{T}$ $n_1=n_2$

Conclusion

From the analysis, we find that the moles of each gas are equal (n_1 = n_2) under the same pressure, volume, and temperature conditions. This is true because of the Ideal Gas Law, which dictates the relationship between the pressure, volume, temperature, and moles of a gas. In this case, since pressure, volume, and temperature are kept the same for both gases, their moles must also be equivalent.