The dimensional formula for entropy is

• A. [ML{T^{ - 2}}{K^1}]
• B. [M{L^0}{T^{ - 2}}]
• C. [M{L^2}{T^{ - 2}}{K^{ - 1}}]
• D. [M{L^2}{T^{ - 2}}K]

To determine the dimensional formula for entropy, we first need to understand what entropy represents in thermodynamics. Entropy is a measure of the disorder or randomness in a system, and it is often associated with the amount of energy in a physical system that is not available to do work. The standard unit of entropy in the International System of Units (SI) is joules per kelvin (J/K). To find the dimensional formula, we can break down the units involved.

Breaking Down the Units

We know that:

• 1 joule (J) is defined as 1 kilogram meter squared per second squared (kg·m²/s²).
• The unit of temperature in the SI system is kelvin (K).

Expressing Joules in Terms of Base Units

Now, let's express joules in terms of its base dimensions:

• J = kg·m²/s²
• In dimensional terms, this is represented as: [M^1 L^2 T^{-2}]

Combining with Temperature

Since entropy is measured in joules per kelvin, we need to include the dimensional formula for temperature:

• Temperature (K) contributes a dimension of [K^1].

Putting It All Together

Now, we can combine these components to find the dimensional formula for entropy:

• Entropy = Joules/Kelvin = [M^1 L^2 T^{-2}] / [K^1]
• This simplifies to: [M^1 L^2 T^{-2} K^{-1}].

Identifying the Correct Option

Now that we have derived the dimensional formula for entropy as [M^1 L^2 T^{-2} K^{-1}], we can compare this with the options provided:

• A. [ML{T^{ - 2}}{K^1}]
• B. [M{L^0}{T^{ - 2}}]
• C. [M{L^2}{T^{ - 2}}{K^{ - 1}}]
• D. [M{L^2}{T^{ - 2}}K]

The correct choice is C: [M{L^2}{T^{ - 2}}{K^{ - 1}}]. This matches our derived formula perfectly.

Conclusion

In summary, the dimensional formula for entropy is [M^1 L^2 T^{-2} K^{-1}], which corresponds to option C. Understanding these relationships helps clarify how physical quantities interact in thermodynamics and provides a solid foundation for further studies in the field.