What will happen to the volume of a gas if its pressure is doubled and its temperature is reduced to half?

QUESTION #518

Asked by: Lance Boswell Boyle's Law simply describes the relationship between the pressure and volume of an enclosed gas when Temperature remains constant. That relationship, usually expressed as P1V1 = P2V2, just means that the product of pressure x volume remains unchanged as either or both are changed. Since pressure x volume remains constant, for example, doubling the pressure on an enclosed gas will reduce its volume to 1/2 its previous size. Tripling the pressure will reduce its volume to 1/3, and so on. Alternatively, if you double the volume available to an enclosed gas, pressure is halved. The simplest demonstration of Boyle's Law is a hand bicycle pump. By pushing down on the piston, the reduced volume increases the pressure of the air inside so that it is forced into the tire. Because pressure changes will have an affect on temperature (feel the pump after a few seconds of pumping), temperature must be allowed to return to its prior value for Boyle's Law to hold true. Answered by: Paul Walorski, B.A., Part-time Physics/Astronomy Instructor

Boyle's Law is a statement of the relationship between the pressure and volume of gasses. Specifically it states that under isothermic conditions, the product of the pressure and volume remains constant, or P1 x V1 = P2 x V2 where P1 is the pressure before some change, V1 is the volume before the change, P2 and V2 are the new values after the change. Another way of thinking about this law is that the values of pressure and volume are inversely proportional; if one goes up, the other must decrease by the same factor. If you trap gas in a cylinder, and then reduce the internal volume of the cylinder to half its original value, the pressure will double. Why does this happen? If you squeeze those gas molecules from the above example into half the volume, you would expect them to be packed closer together and to slam into the sides of the container more often. The sum of all those little collisions is what we call pressure.

Answered by: Rob Landolfi, Science Teacher, Washington, DC

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#.

The correct answer is letter d. its volume is decreased. It is notable that the gas volume is inversely proportional to its pressure and directly proportional to its temperature. With this key points, we can understand that:

If the pressure of a gas had doubled (increased), the volume of a gas will decrease.
If the temperature is reduced to half (colder), the volume will also reduce.

Thus, the total volume of the gas will decrease.

The Pressure and Volume Relationship

It is known that as the gas pressure increases, the gas volume decreases. It is because the particles of gas are forced to be closer together. We used the word conversely, as the gas pressure decreases, the volume of a gas increases. It is because the particles of gas can now move apart more farther.

The Temperature and Volume Relationship

It is known that gases normally expanding when they are heated. It is because the same amount of substance will now occupy a greater volume. The volume of a gas will likely to increase.

You need to know that:

Conversely proportional means the other way around. If one variable will decrease, the other variable will increase and vice versa.
Directly proportional means the same way around. If one variable will decrease, the other variable will also decrease and vice versa.

Code: 10.22.4.1.