The electrons in a metal that produce electric currents behave approximately as molecules of an ideal gas. The mass of an electron is me = 9.109·10-31 kg. If the temperature of the metal is 287 K, what is the root-mean-square speed of the electrons?
To find the root-mean-square speed of the electrons, we can make use of the equation for the root-mean-square speed of gas molecules, which is given by:
v = sqrt((3 * k * T) / (m))
where:
v is the root-mean-square speed,
k is Boltzmann's constant (k = 1.38·10^-23 J/K),
T is the temperature in Kelvin, and
m is the mass of a gas molecule.
In this case, since we are considering the electrons in the metal to behave like molecules of an ideal gas, we can use the given mass of the electron (me = 9.109·10^-31 kg).
Plugging in the values into the equation, we have:
v = sqrt((3 * (1.38·10^-23 J/K) * (287 K)) / (9.109·10^-31 kg))
Let's calculate it step by step:
1. Multiply Boltzmann's constant (k) by the temperature (T):
(1.38·10^-23 J/K) * (287 K) = 3.96306·10^-21 J
2. Multiply the result by 3:
3 * (3.96306·10^-21 J) = 1.18892·10^-20 J
3. Divide by the electron mass (m):
(1.18892·10^-20 J) / (9.109·10^-31 kg) = 1.30594·10^10 m^2/s^2
4. Take the square root of the result:
sqrt(1.30594·10^10 m^2/s^2) ≈ 3.614·10^5 m/s
Therefore, the root-mean-square speed of the electrons in the metal at a temperature of 287 K is approximately 3.614·10^5 m/s.