Let's tackle these questions step by step, starting with the first one regarding the temperature difference between two chunks of metal connected by a rod. This problem involves heat transfer and can be analyzed using the principles of thermal conduction and heat capacity.

Temperature Difference Between Two Metal Chunks

We have two metal chunks with heat capacities C1 and C2, connected by a rod of length L and cross-sectional area S. The rod has a low thermal conductivity k, and the entire system is thermally insulated. Initially, there is a temperature difference ΔT0 between the two metal chunks.

To find the temperature difference as a function of time, we can use the concept of heat transfer through conduction. The rate of heat transfer (Q) through the rod can be described by Fourier's law of heat conduction:

Q = -k * A * (dT/dx)

Where:

• Q is the heat transfer rate
• k is the thermal conductivity of the rod
• A is the cross-sectional area of the rod (S)
• dT/dx is the temperature gradient along the rod

Given that the heat capacity of the rod is negligible, we can assume that the heat lost by one chunk equals the heat gained by the other. The heat transferred can also be expressed in terms of the heat capacities:

Q = C1 * dT1 = C2 * dT2

Where dT1 and dT2 are the changes in temperature of the two metal chunks. Since the system is insulated, we can set up the following differential equation:

m1 * C1 * (dT1/dt) = -m2 * C2 * (dT2/dt)

By integrating this equation, we can find the temperature difference as a function of time. The solution will yield an exponential decay of the temperature difference over time, which can be expressed as:

ΔT(t) = ΔT0 * e^(-kt/(C1 + C2))

This equation shows that the temperature difference decreases exponentially with time, where k is the thermal conductivity of the rod, and C1 and C2 are the heat capacities of the two metal chunks.

Electric Current in a Wire

Next, let's consider the scenario where a constant electric current flows through a uniform wire. The wire has a cross-sectional radius R and a thermal conductivity coefficient k. We want to find the temperature difference across the wire, given that the steady-state temperature at the surface is T0.

The power generated per unit volume in the wire due to the electric current can be expressed as:

P = I^2 / R

Where I is the current and R is the resistance. The heat generated will cause a temperature gradient along the wire, which we can analyze using Fourier's law again. The temperature difference ΔT across the wire can be found using:

ΔT = (P * L) / (k * A)

Substituting the power expression, we can derive:

ΔT = (I^2 * L) / (k * πR^2)

This equation gives us the temperature difference across the wire based on the current flowing through it and its physical properties.

Mixture of Gases

Now, let's move on to the mixture of gases. We have one mole of a monoatomic gas (γ = 5/3) mixed with one mole of a diatomic gas (γ = 7/5). To find the value of γ for the mixture, we can use the formula for the effective heat capacity ratio of a mixture:

γ_mixture = (n1 * γ1 + n2 * γ2) / (n1 + n2)

Substituting the values:

γ_mixture = (1 * (5/3) + 1 * (7/5)) / (1 + 1)

Calculating this gives:

γ_mixture = (5/3 + 7/5) / 2 = (25/15 + 21/15) / 2 = (46/15) / 2 = 23/15

Mixing Two Perfect Gases

For the next question, we are mixing two perfect gases at absolute temperatures T1 and T2, with molecular masses m1 and m2, and number of molecules n1 and n2 respectively. To find the temperature of the mixture, we can use the principle of conservation of energy:

n1 * k * T1 + n2 * k * T2 = (n1 + n2) * k * T_mixture

Where k is the Boltzmann constant. Rearranging gives us:

T_mixture = (n1 * T1 + n2 * T2) / (n1 + n2)

This equation shows that the temperature of the mixture is a weighted average of the temperatures of the two gases, based on the number of molecules.

Temperature of Junctions in Rods

In the scenario with three rods of materials X and Y, we need to calculate the temperatures at junctions B, C, and D when the ends A and E are maintained at 60°C and 10°C, respectively. Given the thermal conductivities k for X (0.92) and Y (0.46), we can apply the concept of thermal resistance.

The heat flow through each rod can be expressed as:

Q = (T_A - T_B) / R_X = (T_B - T_C) / R_Y = (T_C - T_D) / R_X = (T_D - T_E) / R_Y

Where R_X and R_Y are the thermal resistances of the respective materials. By setting up these equations and solving them simultaneously, we can find the temperatures at junctions B, C, and D.

Mercury Level in a U-Tube

Lastly, we have a U-tube containing mercury, where one side is closed and the other is connected to a vacuum pump. When the temperature of the air in the closed tube is raised from 27°C to 111°C, we need to determine how far the mercury level falls on the left side. This can be analyzed using the ideal gas law and the concept of pressure change due to