To understand the displacement of mercury when the temperature of the smaller sphere is increased, we need to consider how the ideal gas law and the principles of thermodynamics apply to this system. Let's break it down step by step.
Understanding the System
We have two glass spheres connected by a tube containing mercury. The smaller sphere has a radius of R, while the larger sphere has a radius of 2R. Initially, both spheres are at the same temperature T, and the system is in equilibrium. When we increase the temperature of the smaller sphere by a small amount t, we need to determine how this affects the pressure in both spheres and subsequently the mercury level in the tube.
Applying the Ideal Gas Law
The ideal gas law states that:
- P = Pressure
- V = Volume
- n = Number of moles of gas
- R = Ideal gas constant
- T = Temperature
For an ideal gas, we can express the pressure in each sphere as:
P = nRT/V
Calculating the Volumes
The volumes of the spheres can be calculated using the formula for the volume of a sphere:
V = (4/3)πr³
For the smaller sphere (radius R):
V₁ = (4/3)πR³
For the larger sphere (radius 2R):
V₂ = (4/3)π(2R)³ = (4/3)π(8R³) = (32/3)πR³
Effect of Temperature Increase
When the temperature of the smaller sphere is increased by t, the new temperature becomes T + t. The pressure in the smaller sphere can be expressed as:
P₁ = nR(T + t)/V₁
Since the number of moles (n) and the gas constant (R) remain unchanged, we can analyze the change in pressure:
P₁ = nR(T + t)/(4/3)πR³
This results in:
P₁ = (3nR/(4πR³))(T + t)
Pressure in the Larger Sphere
The pressure in the larger sphere remains unchanged because its temperature is constant at T:
P₂ = nRT/V₂ = nRT/(32/3)πR³ = (3nRT)/(32πR³)
Pressure Difference and Mercury Displacement
The difference in pressure between the two spheres will cause the mercury to be displaced. The pressure difference (ΔP) can be expressed as:
ΔP = P₁ - P₂
Substituting the expressions for P₁ and P₂, we get:
ΔP = (3nR/(4πR³))(T + t) - (3nRT)/(32πR³)
To find the displacement of mercury, we can relate this pressure difference to the height (h) of the mercury column using the hydrostatic pressure equation:
ΔP = ρgh
Where ρ is the density of mercury and g is the acceleration due to gravity. Rearranging gives us:
h = ΔP/(ρg)
Final Calculation
By substituting the expression for ΔP into the equation for h, we can find the displacement of the mercury. The final result will depend on the specific values of n, R, T, t, ρ, and g, but the key takeaway is that the displacement is directly proportional to the increase in temperature of the smaller sphere and inversely proportional to the density of mercury.
In summary, the increase in temperature of the smaller sphere leads to an increase in pressure, which causes the mercury to be displaced towards the larger sphere. The exact amount of displacement can be calculated using the derived formulas, taking into account the properties of the gases and the mercury involved in the system.
