The pressure of 4.2 L of nitrogen gas in a flexible container is decreased to one-half its original pressure, and its absolute temperature is increased to double the original temperature. What is its volume now?
1 Answer
Explanation:
Your tool of choice for this problem will be the combined gas law equation, which looks like this
color(blue)((P_1V_1)/T_1 = (P_2V_2)/T_2)" "P1V1T1=P2V2T2 , where
Now, even without doing any calculations, you can examine the above equation and predict what you expect the volume of the gas to be at the final state.
Notice that pressure and volume have an inverse relationship when temperature is kept constant - this is known as Boyle's Law.
This tells you that a decrease in pressure will cause an increase in volume.
On the other hand, temperature and volume have a direct relationship when pressure is kept constant - this is known as Charles' Law.
This tells you that an increase in temperature will cause an increase in volume.
In your case, these two effects will actually combine. Increasing the temperature of the gas while decreasing its pressure will result in an increase in volume, regardless of the actual changes in temperature and pressure.
Mathematically, you can prove this by rearranging the combined gas law equation to solve for
V_2 = P_1/P_2 * T_2/T_1 * V_1V2=P1P2⋅T2T1⋅V1
You know that
P_2 = P_1/2 ->P2=P12→ the pressure is halved**T_2 = 2 * T_1 ->T2=2⋅T1→ the temperature is doubled**
This means that you have
V_2 = color(red)(cancel(color(black)(P_1)))/(1/2 * color(red)(cancel(color(black)(P_1)))) * (2 * color(red)(cancel(color(black)(T_1))))/color(red)(cancel(color(black)(T_1))) * V_1
V_2 = 2 * 2 * V_1 = 4 * V_1
Indeed, both effects combine to cause an increase in volume.
The volume of the gas will thus be
V_2 = 4 * "4.2 L" = "16.8 L"
Rounded to two sig figs, the answer will be
V_2 = color(green)("17 L")
