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To find the dimensions of the constants 'a' and 'b' in Van-derwaal's equation, [tex]\(\left( P +\frac{ a }{ V ^2}\right)( V - b )= RT\)[/tex], we need to analyze the dimensions of each term in the equation. The equation involves pressure (P), volume (V), temperature (T), and the gas constant (R).
Let's start by summarizing the dimensions of the given quantities:
1. Pressure, [tex]\(P\)[/tex]:
Pressure is defined as force per unit area. In terms of fundamental units:
[tex]\[ [P] = \frac{[M][L][T^{-2}]}{[L^2]} = [M][L^{-1}][T^{-2}] \][/tex]
2. Volume, [tex]\(V\)[/tex]:
Volume is a measure of space and has the dimensions:
[tex]\[ [V] = [L^3] \][/tex]
3. Temperature, [tex]\(T\)[/tex]:
Temperature is a fundamental quantity and its dimension is simply:
[tex]\[ [T] = [\Theta] \][/tex]
4. Gas constant, [tex]\(R\)[/tex]:
The gas constant has the dimensions given by:
[tex]\[ [R] = \frac{[\text{Energy}]}{[T] \cdot [\text{Amount of substance}]} \][/tex]
Energy (Joule) has the dimension [tex]\([M][L^2][T^{-2}]\)[/tex], thus:
[tex]\[ [R] = \frac{[M][L^2][T^{-2}]}{[T]} = [M][L^2][T^{-2}][\Theta^{-1}] \][/tex]
### Step-by-Step Solution:
1. Analyzing [tex]\((P + \frac{a}{V^2})\)[/tex]:
The term [tex]\(P\)[/tex] is a pressure, so its dimensions are:
[tex]\[ [P] = [M][L^{-1}][T^{-2}] \][/tex]
The term [tex]\(\frac{a}{V^2}\)[/tex] must have the same dimensions as [tex]\(P\)[/tex] because they are added together. The dimensions of [tex]\(V^2\)[/tex] are:
[tex]\[ [V^2] = [L^3]^2 = [L^6] \][/tex]
Therefore, the dimensions of [tex]\(\frac{a}{V^2}\)[/tex] should match the dimensions of [tex]\(P\)[/tex]:
[tex]\[ \left[\frac{a}{V^2}\right] = [M][L^{-1}][T^{-2}] \][/tex]
Rearranging to solve for the dimensions of [tex]\(a\)[/tex]:
[tex]\[ [a] = [M][L^{-1}][T^{-2}] \cdot [L^6] = [M][L^5][T^{-2}] \][/tex]
2. Analyzing [tex]\((V - b)\)[/tex]:
The term [tex]\(V\)[/tex] is volume, so its dimensions are:
[tex]\[ [V] = [L^3] \][/tex]
The term [tex]\(b\)[/tex] must have the same dimensions as [tex]\(V\)[/tex] since they are subtracted:
[tex]\[ [b] = [L^3] \][/tex]
Thus, the dimensions of the constants are:
- The dimension of [tex]\(a\)[/tex] is [tex]\([M][L^5][T^{-2}]\)[/tex],
- The dimension of [tex]\(b\)[/tex] is [tex]\([L^3]\)[/tex].
These dimensions ensure that the units on both sides of the Van-derwaal's equation match appropriately.
Let's start by summarizing the dimensions of the given quantities:
1. Pressure, [tex]\(P\)[/tex]:
Pressure is defined as force per unit area. In terms of fundamental units:
[tex]\[ [P] = \frac{[M][L][T^{-2}]}{[L^2]} = [M][L^{-1}][T^{-2}] \][/tex]
2. Volume, [tex]\(V\)[/tex]:
Volume is a measure of space and has the dimensions:
[tex]\[ [V] = [L^3] \][/tex]
3. Temperature, [tex]\(T\)[/tex]:
Temperature is a fundamental quantity and its dimension is simply:
[tex]\[ [T] = [\Theta] \][/tex]
4. Gas constant, [tex]\(R\)[/tex]:
The gas constant has the dimensions given by:
[tex]\[ [R] = \frac{[\text{Energy}]}{[T] \cdot [\text{Amount of substance}]} \][/tex]
Energy (Joule) has the dimension [tex]\([M][L^2][T^{-2}]\)[/tex], thus:
[tex]\[ [R] = \frac{[M][L^2][T^{-2}]}{[T]} = [M][L^2][T^{-2}][\Theta^{-1}] \][/tex]
### Step-by-Step Solution:
1. Analyzing [tex]\((P + \frac{a}{V^2})\)[/tex]:
The term [tex]\(P\)[/tex] is a pressure, so its dimensions are:
[tex]\[ [P] = [M][L^{-1}][T^{-2}] \][/tex]
The term [tex]\(\frac{a}{V^2}\)[/tex] must have the same dimensions as [tex]\(P\)[/tex] because they are added together. The dimensions of [tex]\(V^2\)[/tex] are:
[tex]\[ [V^2] = [L^3]^2 = [L^6] \][/tex]
Therefore, the dimensions of [tex]\(\frac{a}{V^2}\)[/tex] should match the dimensions of [tex]\(P\)[/tex]:
[tex]\[ \left[\frac{a}{V^2}\right] = [M][L^{-1}][T^{-2}] \][/tex]
Rearranging to solve for the dimensions of [tex]\(a\)[/tex]:
[tex]\[ [a] = [M][L^{-1}][T^{-2}] \cdot [L^6] = [M][L^5][T^{-2}] \][/tex]
2. Analyzing [tex]\((V - b)\)[/tex]:
The term [tex]\(V\)[/tex] is volume, so its dimensions are:
[tex]\[ [V] = [L^3] \][/tex]
The term [tex]\(b\)[/tex] must have the same dimensions as [tex]\(V\)[/tex] since they are subtracted:
[tex]\[ [b] = [L^3] \][/tex]
Thus, the dimensions of the constants are:
- The dimension of [tex]\(a\)[/tex] is [tex]\([M][L^5][T^{-2}]\)[/tex],
- The dimension of [tex]\(b\)[/tex] is [tex]\([L^3]\)[/tex].
These dimensions ensure that the units on both sides of the Van-derwaal's equation match appropriately.
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