Name: Molar Specific Heat Capacities of Gases
Uploaded: 2019-07-19
Duration: 10 min 31 s
Description: Do all gases have the same molar heat capacities? Are there certain criteria for determining molar specific heat capacities of gases? Watch the video to find...

Question

A piece of metal floats on mercury. The coefficient of volume expansion of metal and mercury are

γ_{1}

and

γ_{2}

respectively. If the temperature of both mercury and metal are increased by an amount

Δ T

, by what factor does the fraction of the volume of the metal submerged in mercury changes?

Accepted Answer

On increasing temperature, the density $(ρ)$ of material decreases.

The relation between density and the coefficient $(γ)$ of volumetric expansion and increase in temperature $(∆ T)$ is $ρ' = ρ (1 - γ ∆ T)$

Let $V$ be the total volume and $V_{l}$ be the volume of metal immersed in mercury.

By using archimedes principle $ρ_{m e t a l} \times V \times g = ρ_{m e r c u r y} \times V_{l} \times g$

The fraction of the volume immersed inside the mercury is given by, $f = \frac{V_{l}}{V} = \frac{ρ_{metal}}{ρ_{mercury}}$

On the increasing the temperature, the new fraction of the volume immersed in the mercury will be

$f' = \frac{ρ'_{metal}}{ρ'_{mercury}}$

$= \frac{ρ_{metal} (1 - γ_{1} ∆ T)}{ρ_{mercury} (1 - γ_{2} ∆ T)}$

The ratio of the fractions,

$\Rightarrow \frac{f'}{f} = \frac{ρ_{metal} (1 - γ_{1} ∆ T)}{ρ_{mercury} (1 - γ_{2} ∆ T)} \times \frac{ρ_{mercury}}{ρ_{metal}}$

$= \frac{(1 - γ_{1} ∆ T)}{(1 - γ_{2} ∆ T)}$

$= (1 - γ_{1} ∆ T) {(1 - γ_{2} ∆ T)}^{- 1}$

$= (1 - γ_{1} ∆ T) (1 + γ_{2} ∆ T)$

$∵$ As ${(1 - γ_{2} ∆ T)}^{- 1} \approx (1 + γ_{2} ∆ T)$

$= 1 + γ_{2} ∆ T - γ_{1} ∆ T - γ_{1} γ_{2} {(∆ T)}^{2}$

$= 1 + (γ_{2} - γ_{1}) (∆ T)$

$∵ γ_{1} γ_{2} {(∆ T)}^{2}$ is negligible.

So the fractional change in volume,

$\Rightarrow \frac{f'}{f} - 1 = \frac{f' - f}{f} = (γ_{2} - γ_{1}) (∆ T)$

Important Questions on Thermometry, Thermal Expansion and Kinetic Theory of Gases

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A piece of metal floats on mercury. The coefficient of volume expansion of metal and mercury are γ1 and γ2 respectively. If the temperature of both mercury and metal are increased by an amount ΔT, by what factor does the fraction of the volume of the metal submerged in mercury changes?

Important Questions on Thermometry, Thermal Expansion and Kinetic Theory of Gases

Learn and Prepare for any exam you want !