For an ideal gas, for the pressure and volume of the gas to remain constant, if the number of moles of the gas is increased by a factor of four, by what factor must the temperature of the gas change?
Here, we’re considering an ideal gas. This means the gas is described by the ideal gas law. The ideal gas law tells us that the pressure of an ideal gas multiplied by its volume equals the number of moles of the gas multiplied by a constant times the gas’s temperature. In our application of this law, we know that as always 𝑅 is a constant value. This is called the molar gas constant. In addition to this, though, the pressure and volume of our gas, represented by 𝑃 and 𝑉, are to remain constant too.
Everything in this equation then, except for the number of moles 𝑛 and the temperature of the gas 𝑇, are constant values. We can even move all of the constant values onto the same side of the equation by dividing both sides of the equation by the molar gas constant 𝑅. That causes this factor to cancel on the right, and we find that 𝑃 times 𝑉 divided by 𝑅 is equal to 𝑛 times 𝑇. Since 𝑃 and 𝑉 and 𝑅 are all constants individually, we know that 𝑃 times 𝑉 divided by 𝑅 is itself a constant. Let’s call this constant capital 𝐶. Switching the sides of our equation, we have then that 𝑛 times 𝑇 is equal to a constant that we’ve called capital 𝐶.
At this point, we’re told to imagine that the number of moles of the gas is increased by a factor of four. In other words, 𝑛 becomes four times 𝑛. This new number of moles multiplied by a new temperature must still equal the constant 𝐶. Since we have now what we can call an extra factor of four in this equation, that means our temperature must change in such a way that this factor of four is effectively neutralized or cancelled out. We know that four times one-fourth is one, and therefore this is the factor by which the temperature of the gas must change.
If the number of moles of the gas is increased by a factor of four, while pressure and volume are held constant, then the temperature of the gas must be reduced by a factor of four. That is, we multiply the temperature by one-fourth.