Energy density
Van der Waals Equation Dimensional Analysis
The Van der Waals equation for real gases is:
$ \left(P+\frac{a}{V^2}\right) (V-b)=RT $
For the equation to be dimensionally valid, terms added or subtracted must have identical dimensions.
Dimensions of Constant 'b'
From the term $(V-b)$, constant $b$ must have the dimensions of Volume ($V$).
- Dimension of Volume ($V$): $[L^3]$
- Dimension of $b$: $[L^3]$
Dimensions of Constant 'a'
From the term $\left(P+\frac{a}{V^2}\right)$, the term $\frac{a}{V^2}$ must have the dimensions of Pressure ($P$).
- Dimension of Pressure ($P$): $[ML^{-1}T^{-2}]$
- Dimension of $V^2$: $([L^3])^2 = [L^6]$
- Dimensions of $a$ = Dimensions of $\left(\frac{a}{V^2}\right) \times$ Dimensions of $V^2$
- Dimensions of $a$ = $[ML^{-1}T^{-2}] \times [L^6] = [ML^5T^{-2}]$
Calculating Dimensions of $ab^{-2}$
The dimensions of the term $a \cdot b^{-2}$ are derived as follows:
- Dimension of $a \cdot b^{-2}$ = $\frac{\text{Dimension of } a}{(\text{Dimension of } b)^2}$
- Dimension of $a \cdot b^{-2}$ = $\frac{[ML^5T^{-2}]}{[L^3]^2}$
- Dimension of $a \cdot b^{-2}$ = $\frac{[ML^5T^{-2}]}{[L^6]}$
- Dimension of $a \cdot b^{-2}$ = $[ML^{-1}T^{-2}]$
Comparing Dimensions with Options
The calculated dimensions $[ML^{-1}T^{-2}]$ represent Pressure. Let's examine the dimensions of the given options:
| Option | Physical Quantity | Dimensions |
|---|---|---|
| 1 | Compressibility | $[M^{-1}LT]$ |
| 2 | Energy density | $[ML^{-1}T^{-2}]$ |
| 3 | Planck's constant | $[ML^2T^{-1}]$ |
| 4 | Strain | Dimensionless |
The dimensions of $ab^{-2}$ are equivalent to those of Pressure and Energy density.
Similar Questions
A person measures mass of 3 different particles as 435.42 g, 226.3 g and 0.125 g. According to the rules for arithmetic operations with significant figures, the addition of the masses of 3 particles will be.
Match the LIST-I with LIST-II
LIST-I LIST-II A. Gravitational constant I. $[LT^{-2}]$ B. Gravitational potential energy II. $[L^2T^{-2}]$ C. Gravitational potential III. $[ML^2T^{-2}]$ D. Acceleration due to gravity IV. $[M^{-1}L^3T^{-2}]$ Choose the correct answer from the options given below:
A particle is released from height S above the surface of the earth. At certain height its kinetic energy is three times its potential energy. The height from the surface of the earth and the speed of the particle at that instant are respectively.
The angle of projection of a particle is measured from the vertical axis as $\phi$ and the maximum height reached by the particle is $h_m$. Here $h_m$ as function of $\phi$ can be presented as
Which of the following curves possibly represent one-dimensional motion of a particle?
A force of 49 N acts tangentially at the highest point of a sphere (solid) of mass 20 kg, kept on a rough horizontal plane. If the sphere rolls without slipping, then the acceleration of the center of the sphere is
Consider a cylindrical tank completely filled with water of height $1.6 \ m$ and cross-sectional area $0.5 \ m^2$. There is a hole in its side at a height of $90 \ cm$ from the bottom. Assume the cross-sectional area of the hole to be negligibly small compared to the cross-sectional area of the water tank. If a load of $50 \ kg$ is applied on the upper surface of water in the tank, then at the moment the hole is opened, the velocity of water coming out is:
($g = 10 \ m/s^2$)
Three identical spheres of mass m, are placed at the vertices of an equilateral triangle of length a. When released, they interact only through gravitational force and collide after a time T = 4 seconds. If the sides of the triangle are increased to length 2a and also the masses of the spheres are made 2m, then they will collide after __________ seconds.
A 4.0 cm long straight wire carrying a current of 8A is placed perpendicular to a uniform magnetic field of strength 0.15 T. The magnetic force on the wire is __________ mN.
- Match List - I with List - II.Choose the correct answer from the options given below :
List - I List - II (A) Coefficient of viscosity (I) $[ML^0T^{-3}]$ (B) Intensity of wave (II) $[ML^{-2}T^{-2}]$ (C) Pressure gradient (III) $[M^{-1}LT^2]$ (D) Compressibility (IV) $[ML^{-1}T^{-1}]$
Important Questions from Mechanics
A person measures mass of 3 different particles as 435.42 g, 226.3 g and 0.125 g. According to the rules for arithmetic operations with significant figures, the addition of the masses of 3 particles will be.
Match the LIST-I with LIST-II
LIST-I LIST-II A. Gravitational constant I. $[LT^{-2}]$ B. Gravitational potential energy II. $[L^2T^{-2}]$ C. Gravitational potential III. $[ML^2T^{-2}]$ D. Acceleration due to gravity IV. $[M^{-1}L^3T^{-2}]$ Choose the correct answer from the options given below:
A particle is released from height S above the surface of the earth. At certain height its kinetic energy is three times its potential energy. The height from the surface of the earth and the speed of the particle at that instant are respectively.
The angle of projection of a particle is measured from the vertical axis as $\phi$ and the maximum height reached by the particle is $h_m$. Here $h_m$ as function of $\phi$ can be presented as
Which of the following curves possibly represent one-dimensional motion of a particle?
