# When the pressure of a gas is halved and its temperature is doubled, what happens to its volume?

##### 1 Answer

The volume of the gas will increase by a factor of

#### Explanation:

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 **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 **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 -># thepressureof the gas ishalved

#T_2 = 2 * T_1 -># thetemperatureof the gas isdoubled

Rearrange the combined gas law equation to solve for

#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.