Your calculator can't handle enormous exponents, but we can make sense of large powers of e by converting them to large powers of 10. If we write e = 10^α, then e^β = (10^α)^β = 10^αβ. b. What is the multiplicity of a macrostate with entropy S = 1.0 J/K? Give your answer as a power of 10.

Verified step by step guidance

1. First, understand that the multiplicity of a macrostate, denoted as \( \Omega \), is related to the entropy, \( S \), by the Boltzmann's entropy formula: \( S = k \ln \Omega \), where \( k \) is the Boltzmann constant (approximately \( 1.38 \times 10^{-23} \, \text{J/K} \)).
2. Rearrange the formula to solve for \( \Omega \): \( \Omega = e^{S/k} \).
3. Substitute the given value of entropy, \( S = 1.0 \, \text{J/K} \), into the equation: \( \Omega = e^{1.0 / 1.38 \times 10^{-23}} \).
4. To simplify the calculation, use the approximation \( e = 10^{\alpha} \) where \( \alpha \approx 0.434 \). This converts the expression to a power of 10: \( \Omega = 10^{(1.0 / (1.38 \times 10^{-23} \times 0.434))} \).
5. Calculate the exponent to express \( \Omega \) as a power of 10, which will give the multiplicity of the macrostate in a manageable form.

Key Concepts

Entropy

Entropy is a measure of the disorder or randomness in a system, often associated with the number of microscopic configurations that correspond to a macroscopic state. In thermodynamics, it quantifies the amount of energy in a physical system that is not available to do work. The higher the entropy, the greater the number of possible microstates, leading to a more disordered system.

Multiplicity

Multiplicity refers to the number of different ways a particular macrostate can be realized by its microstates. It is a crucial concept in statistical mechanics, where the multiplicity of a macrostate is directly related to its entropy. A higher multiplicity indicates a greater number of configurations, which corresponds to higher entropy.

Exponential Growth and Logarithms

Exponential growth describes a process where a quantity increases at a rate proportional to its current value, often represented as e^x or 10^x. Logarithms are the inverse operations of exponentiation, allowing us to express large numbers in a more manageable form. In the context of entropy and multiplicity, converting between bases (like e and 10) helps simplify calculations involving large powers.