Question

Find the dimensions of (a) the specific heat capacity c, (b) the coefficient of linear expansion a and (c) the gas constant R.

Concept of Physics - 1 , HC VERMA , Chapter "Introduction to Physics".

Answer 1

Hello Dear.



(a) ∵ Heat Energy(Q) = m × c × Δt
where, m = mass of the body, c = specific heat capacity and Δt is the change in temperature.

∴ c = Q/mΔt

Unit of Heat = Joule.
= N × m.
= kg ms⁻² × m
= kg m² s⁻²

∴ Dimension of Q = M L² T⁻²

Now, Let us finding the Dimensions of Specific Heat Capacity.

∴ [c] = (M L² T⁻²) ÷ (M K)
[Since, the Dimension of Temperature is K]
  [c] = L² T⁻² K⁻¹

Hence, the Dimensions of Specific Heat Capacity c is L² T⁻² K⁻¹.

________________________-


(b) ∵ α = (L₁ - L₂)/L₀(T₂ -T₁)
∴ Dimension of α = L/LK
  = K⁻¹

Hence, the Dimension of Coefficient of Linear Expansion is K⁻¹.

_________________________


(c) ∵ PV = nRT
where, R is the gas constant,n is the number of moles in the sample of gas,P is the Pressure, V is the Volume of the Gas, & T is the Temperature.

∴ R = PV/nT

For the Dimension of Pressure,

∵ Pressure = Force/Area.
    = Mass × Acceleration/Area
    = M × L T⁻²/L²
    = M L⁻¹ T⁻²

Dimension of Volume = L³
[∵ Volume = (Length)³ ]

Dimension of Temperature = K
Dimension of n = mol.

Now, 
Dimension of Gas Constant R = [(M L⁻¹ T⁻²) × (L³)] ÷ (mol)K
= M L² T⁻² K⁻¹ mol¹⁻.

Hence, the Dimension of the Gas Constant R is M L² T⁻² K⁻¹ mol¹⁻.


_______________________


Hope it helps.

Answer 2

Answer:

Explanation:

(a) ∵ Heat Energy(Q) = m × c × Δt

where, m = mass of the body, c = specific heat capacity and Δt is the change in temperature.

∴ c = Q/mΔt

Unit of Heat = Joule.

= N × m.

= kg ms⁻² × m

= kg m² s⁻²

∴ Dimension of Q = M L² T⁻²

Now, Let us finding the Dimensions of Specific Heat Capacity.

∴ [c] = (M L² T⁻²) ÷ (M K)

[Since, the Dimension of Temperature is K]

  [c] = L² T⁻² K⁻¹

Hence, the Dimensions of Specific Heat Capacity c is L² T⁻² K⁻¹.

________________________-

(b) ∵ α = (L₁ - L₂)/L₀(T₂ -T₁)

∴ Dimension of α = L/LK

  = K⁻¹

Hence, the Dimension of Coefficient of Linear Expansion is K⁻¹.

_________________________

(c) ∵ PV = nRT

where, R is the gas constant,n is the number of moles in the sample of gas,P is the Pressure, V is the Volume of the Gas, & T is the Temperature.

∴ R = PV/nT

For the Dimension of Pressure,

∵ Pressure = Force/Area.

    = Mass × Acceleration/Area

    = M × L T⁻²/L²

    = M L⁻¹ T⁻²

Dimension of Volume = L³

[∵ Volume = (Length)³ ]

Dimension of Temperature = K

Dimension of n = mol.

Now, 

Dimension of Gas Constant R = [(M L⁻¹ T⁻²) × (L³)] ÷ (mol)K

= M L² T⁻² K⁻¹ mol¹⁻.

Hence, the Dimension of the Gas Constant R is M L² T⁻² K⁻¹ mol¹⁻.

_______________________