# Fifty liters of gas is kept at a temperature of 200 K and under pressure of 15 atm. The temperature of the gas is increased to 400 K. The pressure is decreased to 7.5 atm. What is the resulting volume of the gas?

##### 1 Answer

#### Explanation:

Even without doing any calculations, you should be able to look at the information given and predict that the volume of the gas will **increase** after temperature is *increased* and pressure is *decreased*.

In fact, these changes in pressure and temperature will *"work together"* to produce a more significant increase in volume.

So, you know that the sample starts at a pressure of

Now, what would happen if you were to increase the temperature to **keep the pressure constant?**

As you know, volume and temperature have a **direct relationship** when pressure and number of moles are kept constant **Charles' Law**.

Mathematically, this is written as

#color(blue)(V_1/T_1 = V_2/T_2)" "# , where

Since you're essentially doubling the temperature of the gas, you have

#V_2 = (2color(red)(cancel(color(black)(T_1))))/color(red)(cancel(color(black)(T_1))) * V_1 = 2 V_1#

Next, imagine that you're back to the original sample of gas. What would happen if you were to decrease the pressure to **keep the temperature constant?**.

When temperature and number of moles are kept constant, pressure and volume have an **inverse relationship** **Boyle's Law**.

Mathematically, you can write this as

#color(blue)(P_1V_1 = P_2V_2)#

In your case, the pressure is halved, so you have

This will give you

#V_2 = (color(red)(cancel(color(black)(P_1))))/(1/2color(red)(cancel(color(black)(P_1)))) * V_1 = 2V_1#

Once again, the volume is doubled. This means that if you double the temperature **and** halve the pressure, the volume will increase by a factor of

This is what the **combined gas law** equation is all about

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

Rearrange to solve for

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

Plug in your values to confirm that the volume increases by a factor of

#V_2 = (15 color(red)(cancel(color(black)("atm"))))/(7.5color(red)(cancel(color(black)("atm")))) * (400color(red)(cancel(color(black)("K"))))/(200color(red)(cancel(color(black)("K")))) * "50 L" = color(green)("200 L")#