An ideal gas undergoes a transformation in which both its pressure and volume double. How many times does the root mean square speed of the gas molecules increase?

In solving this, the ideal gas law must be considered (pV=nRT). Since both the pressure and volume of the gas have doubled, the product p*V (pressure times volume) has increased four times from the initial to the final state. Also, the quantity of gas was not modified during the process, so the only quantity in the right hand side of the state equation that changes is the temperature. Thus, the temperature of the gas must have increased four times for the equation to hold. Finally, the root mean square (rms) speed of the gas molecules can be written in terms of Boltzmann's constant and the mass of a gas molecule (which are constants) multiplied by the square root of the gas' absolute temperature (so the rms speed is directly proportional to the square root of the temperature). Thus, since the temperature of the gas has increased four times, we can conclude that the rms speed has increased by a factor of 4^(1/2), which is 2.