Question

In the equation \(\bigg[X + {\frac{a}{Y^2}}\bigg] [Y - b] = RT\), X is pressure, Y is volume, R is universal gas constant and T is temperature. The physical quantity equivalent to the ratio \(\frac{a}{b}\) is 

• A. Energy
• B. Pressure gradient
• C. Impulse
• D. Coefficient of viscosity

Answer 1

The Correct Option is A

To determine the physical quantity equivalent to the ratio \(\frac{a}{b}\) in the given equation:

\[ \bigg[X + \frac{a}{Y^2}\bigg] (Y - b) = RT \]

we need to analyze the dimensions or units of the involved variables.

1. In the equation, \(X\) is pressure, \(Y\) is volume, \(R\) is the universal gas constant, and \(T\) is temperature.
2. Dimensionally, pressure \(X\) has units of Force per Unit Area, typically \([\text{ML}^{-1}\text{T}^{-2}]\).
3. Volume \(Y\) has units of \([\text{L}^{3}]\).
4. The term \(\frac{a}{Y^2}\) must be dimensionally consistent with pressure \(X\) since they are added together. Therefore, \(a\) should have units of \([\text{ML}^{5}\text{T}^{-2}]\).
5. The term \((Y - b)\) involves volume \(Y\) so \(b\) must have dimensions of volume, \([\text{L}^{3}]\).
6. Thus, the ratio \(\frac{a}{b}\) will have units: \[ \frac{[\text{ML}^{5}\text{T}^{-2}]}{[\text{L}^3]} = [\text{ML}^{2}\text{T}^{-2}] \] which corresponds to the units of Energy.

By analyzing the dimensions, we conclude that the physical quantity equivalent to the ratio \(\frac{a}{b}\) is Energy.

We can rule out other options because:

• Pressure gradient: has units \([\text{ML}^{-3}\text{T}^{-2}]\), which is inconsistent with \([\text{ML}^{2}\text{T}^{-2}]\).
• Impulse: has the same units as momentum, \([\text{MLT}^{-1}]\), different from energy.
• Coefficient of viscosity: has units of \([\text{ML}^{-1}\text{T}^{-1}]\), not matching energy.

Hence, the correct answer is Energy.