In dimensional analysis, can temperature be expressed as a derived quantity in terms of length, mass and time?

Introduction / Context:
Dimensional analysis classifies physical quantities as either fundamental or derived. Fundamental quantities are independent and cannot be expressed in terms of other base quantities. Derived quantities can be expressed as combinations of these base dimensions. This question examines your understanding of where temperature fits in this framework and whether it can be expressed only using length, mass, and time.

Given Data / Assumptions:

• We consider the standard SI system of units.
• Base quantities include length (L), mass (M), time (T), electric current (I), thermodynamic temperature (Θ), amount of substance (N), and luminous intensity (J in some notations).
• The question offers combinations of length, mass and time only.
• No explicit formula relating temperature only to L, M and T is provided.

Concept / Approach:
In the SI system, temperature is a fundamental quantity with its own independent dimension, often denoted by Θ. Quantities like velocity and acceleration are derived from length and time, while force and energy are derived from length, mass and time. However, temperature is not defined in terms of these mechanical quantities. It is associated with the average kinetic energy of particles, but dimensionally it is treated as an independent base quantity. Therefore, temperature cannot be purely expressed as a derived quantity using only length, mass and time.

Step-by-Step Solution:
Step 1: Recall the list of SI base quantities: length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity. Step 2: Note that temperature is itself one of these base quantities (thermodynamic temperature), usually measured in kelvin. Step 3: Recognise that derived quantities have dimensions formed from combinations of base quantities, such as speed (L T^-1) or force (M L T^-2). Step 4: Observe that there is no general dimensional formula that expresses temperature solely as a combination of L, M, and T. Step 5: Therefore, temperature cannot be expressed as a derived quantity in terms of only length, mass and time. Step 6: Conclude that the correct choice is that temperature cannot be expressed in terms of any of the given combinations.

Verification / Alternative check:
In dimensional analysis tables, temperature is denoted independently as [Θ]. When we write the dimensional formula of other quantities like gas constant R or Boltzmann constant k, temperature appears as a base dimension, not something eliminated in favour of L, M and T. This confirms that temperature is not reducible to mechanical dimensions alone. Even though microscopic interpretations relate temperature to average kinetic energy per particle, which includes mass and velocity, the proportionality constants and statistical nature mean that temperature retains its own base dimension in the SI system.

Why Other Options Are Wrong:
Length and mass: No standard dimensional formula expresses temperature only as L and M.
Mass and time: This pair does not capture the concept of temperature and appears in no fundamental definition of it.
Length, mass and time: Although many mechanical quantities use this combination, temperature is not one of them because it is an independent thermodynamic base quantity.

Common Pitfalls:
A common misconception is to think that because temperature is related to average kinetic energy, it must be expressible in terms of M, L and T. However, this confuses physical interpretation with dimensional independence. Dimensional analysis treats temperature as a separate axis in dimensional space. Remember that not all physical ideas that relate conceptually will reduce dimensionally to the same base quantities.

Final Answer:
Temperature cannot be expressed as a derived quantity using only length, mass and time. The correct option is In terms of none of these.