Question

Statistical mechanics says that in a gas maintained at a constant temperature through thermal contact with a bigger system (a "reservoir") at that temperature, the fluctuations in internal energy are typically a fraction 1/√N of the internal energy. As a fraction of the total internal energy of a mole of gas, how big are the fluctuations in the internal energy? Are we justified in ignoring them?

(a) 1/√N
(b) 1/N
(c) 1/N^2
(d) 1/N^3

Answer 1

Statistical mechanics states that the fluctuations in the internal energy of a gas are typically a fraction of 1/√N of the internal energy, where N is the number of particles in the gas. For a mole of gas, the fluctuations are typically 1/√N the size of the internal energy. Whether we can ignore these fluctuations depends on the specific situation.

Step-by-step explanation:

Statistical mechanics states that in a gas maintained at a constant temperature, the fluctuations in internal energy are typically a fraction of 1/√N of the internal energy, where N is the number of particles in the gas. For a mole of gas, which contains approximately 6.022 x 10^23 particles, the fluctuations would be a fraction of 1/√(6.022 x 10^23) of the internal energy. Simplifying this fraction gives us an answer of (a) 1/√N. Therefore, the fluctuations in the internal energy of a mole of gas are typically 1/√N the size of the internal energy.

Whether we can ignore these fluctuations depends on the specific situation. In many cases, the fluctuations are small enough to be considered negligible for practical purposes, and we can justify ignoring them. However, in certain scenarios, such as for small bodies or when studying the behavior of individual particles, the fluctuations can become significant and should be taken into account.