What do you expect to happen to the volume of a gas if its pressure is doubled and its temperature is reduced?

By Serm Murmson

The combined gas law is an equation that relates the pressure, temperature and volume of a gas. Based on this equation, you can predict what will happen to a sample of gas if you change one of the variables. If you decrease pressure and temperature simultaneously, you will not know the exact result unless you know the exact values by which each quantity decreased.

The combined gas law is a combination of Boyle's law and Charles's law. Boyle's Law states that the volume of a gas held at constant temperature is inversely proportional to its pressure. Charles's law states that the volume of a gas held at constant pressure is directly proportional to its temperature in Kelvins. The combined gas law is the combination of these two relationships into one equation.

The combined gas law states that the pressure of a gas is inversely related to the volume and directly related to the temperature. If temperature is held constant, the equation is reduced to Boyle's law. Therefore, if you decrease the pressure of a fixed amount of gas, its volume will increase. However, if you were to maintain a constant volume while decreasing pressure, the temperature would also have to decrease. The relationship of gases at a constant volume is given by Gay-Lussac's law. Gay-Lussac's law states that at constant volume, the pressure and temperature of a gas are directly proportional.

If you decrease the temperature of a fixed amount of gas at constant pressure, the combined gas law reduces to Charles's law. In this case, volume will also decrease. If you were to maintain a constant volume while decreasing the temperature, the pressure would also have to decrease in order to satisfy Gay-Lussac's law.

If you decrease both the pressure and temperature of a fixed amount of gas, any changes you observe will be in the volume of the gas. Temperature is directly related to volume, and pressure is inversely related to volume. Therefore, the resulting changes in volume will depend on the quantitative changes in both pressure and temperature. For example, if you decrease the temperature of the gas by a greater degree than the decrease in pressure, the volume will decrease. However, if you decrease the pressure by a greater degree than the decrease in temperature, the volume will increase.

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Your tool of choice here will be the combined gas law equation, which looks like this

#color(blue)(ul(color(black)((P_1V_1)/T_1 = (P_2V_2)/T_2)))#

Here

#P_1#, #V_1#, #T_1# are the pressure, volume, and absolute temperature of the gas at an initial state

#P_2#, #V_2#, #T_2# are the pressure, volume, and absolute temperature of the gas at a final state

Now, notice that when the temperature is kept constant, increasing the pressure of the gas by a specific factor will cause its volume to decrease by the same factor #-># think Boyle's Law here.

Similarly, when the pressure is kept constant, increasing the temperature of the gas by a specific factor will cause its volume to increase by the same factor #-># think Charles' law here.

This tells you that increasing the temperature of the gas will cause its volume to increase. Similarly, decreasing its pressure will also cause its volume to increase.

So even without doing any calculations, you should be able to say that

#V_2 > V_1#

In other words, decreasing the pressure of the gas and increasing its temperature are changes that do not compete with each other in terms of their influence on the volume of the gas.

In your case, you have

#P_2 = P_1/2 -># the pressure of the gas is halved

#T_2 = 2 * T_1 -># the temperature of the gas is doubled

Rearrange the combined gas law equation to solve for #V_2#

#V_2 = P_1/P_2 * T_2/T_1 * V_1#

Plug in your values to find

#V_2 = overbrace(color(red)(cancel(color(black)(P_1)))/(color(red)(cancel(color(black)(P_1)))/2))^(color(blue)("influence of pressure")) * overbrace( (2 * color(red)(cancel(color(black)(T_1))))/color(red)(cancel(color(black)(T_1))))^(color(blue)("influence of temperature")) * V_1#

#V_2 = 2 * 2 * V_1#

#color(darkgreen)(ul(color(black)(V_2 = 4 * V_1)))#

As predicted, the volume of the gas increased as a result of the two changes. Notice that the volume increased by a factor that is equal to the product of the factor that corresponds to the decrease in pressure, i.e. #2#, and the factor that corresponds to the increase in temperature, i.e. #2#.

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