H2 Physics 12 - Temperature & Ideal Gases: Finding the r.m.s. Speed

Step 1 - State the problem and given data

A container holds zero point zero two four kilograms of nitrogen gas at a temperature of three hundred kelvin.

Step 1 - State the problem and given data

The molar mass of nitrogen is twenty-eight point zero grams per mole.

Step 1 - State the problem and given data

We want to find the root-mean-square speed of the nitrogen molecules at this temperature.

Step 2 - Write the kinetic theory expression for r.m.s. speed

From the kinetic theory of ideal gases, the mean translational kinetic energy of one molecule is three halves k T.

Step 2 - Write the kinetic theory expression for r.m.s. speed

This equals one half m c-r-m-s squared, where m is the mass of one molecule.

Step 2 - Write the kinetic theory expression for r.m.s. speed

Rearranging gives c-r-m-s equals the square root of three k T over m.

Step 3 - Find the mass of one nitrogen molecule

Each nitrogen molecule, N two, has a molar mass of zero point zero two eight zero kilograms per mole.

Step 3 - Find the mass of one nitrogen molecule

The mass of one molecule m equals the molar mass divided by Avogadro's number.

Step 3 - Find the mass of one nitrogen molecule

That gives m equals zero point zero two eight zero divided by six point zero two times ten to the twenty-three, which is four point six five times ten to the minus twenty-six kilograms.

Step 4 - Substitute and compute the r.m.s. speed

Now substitute: c-r-m-s equals the square root of three times one point three eight times ten to the minus twenty-three times three hundred, divided by four point six five times ten to the minus twenty-six.

Step 4 - Substitute and compute the r.m.s. speed

The numerator is one point two four two times ten to the minus twenty.

Step 4 - Substitute and compute the r.m.s. speed

Dividing gives two point six seven times ten to the five.

Step 4 - Substitute and compute the r.m.s. speed

Taking the square root gives approximately five hundred and seventeen metres per second.

Step 5 - Alternative using molar form and key takeaway

An equivalent formula uses the molar gas constant R instead of Boltzmann's constant.

Step 5 - Alternative using molar form and key takeaway

c-r-m-s equals the square root of three R T over M, where M is the molar mass in kilograms per mole.

Step 5 - Alternative using molar form and key takeaway

This gives the same answer and avoids computing the mass of one molecule.

Step 5 - Alternative using molar form and key takeaway

Remember: r-m-s speed is proportional to the square root of temperature in kelvin, so doubling the absolute temperature increases the r-m-s speed by a factor of root two, not two.