A flask of volume V contain some mercury it is found that at different temperature the volume of air inside theflaskremains the same. If gamma g and gamma m are the coefficient of cubical expansion of glass and mercuryrespectively. Find the volvolume of the mercury in the flask .

To find the volume of mercury in the flask, we can consider the volume of air inside the flask as a constant. This means that the change in volume due to temperature changes will only be caused by the expansion or contraction of the glass and mercury.

Let's assume the initial volume of the flask is V0 and the volume of the air inside it is V_air.

When the temperature changes, the volume of the flask Vf will be given by:

Vf = V0 + ΔV_glass + ΔV_mercury

Where ΔV_glass is the change in volume due to the expansion or contraction of the glass and ΔV_mercury is the change in volume due to the expansion or contraction of the mercury.

We know that the coefficient of cubical expansion of the glass is represented by γg and the coefficient of cubical expansion of mercury is represented by γm.

ΔV_glass = γg * V0 * ΔT

Where ΔT is the change in temperature.

ΔV_mercury = γm * V_mercury * ΔT

Where V_mercury is the volume of the mercury in the flask.

Since the volume of air inside the flask remains the same, we can also say:

V_air = V0 - V_mercury

Substituting the expressions for ΔV_glass and ΔV_mercury into the equation for Vf, we get:

Vf = V0 + γg * V0 * ΔT + γm * V_mercury * ΔT

Since Vf = V_air, we can substitute V_air with V0 - V_mercury in the above equation:

V0 - V_mercury = V0 + γg * V0 * ΔT + γm * V_mercury * ΔT

Now, we can solve the equation for V_mercury:

V_mercury = (V0 - V0 * γg * ΔT) / (1 + γm * ΔT)

So, the volume of the mercury in the flask, V_mercury, can be calculated using the given values of V0, γg, γm, and ΔT.

To find the volume of the mercury in the flask, we can use the principle of maintaining constant volume of air inside the flask at different temperatures.

Let's assume that the initial volume of the mercury is Vm and the initial volume of the air is Va.

At a different temperature, the volume of the mercury expands to Vm' and the volume of the air remains the same, Va.

According to the principle of maintaining constant volume:

V = Vm + Va

At temperature T, for the mercury:

Vm' = Vm * (1 + γm * ΔT)

Where γm is the coefficient of cubical expansion of mercury and ΔT is the change in temperature.

For the glass:

Va = Va * (1 + γg * ΔT)

Where γg is the coefficient of cubical expansion of glass and ΔT is the change in temperature.

Since Va remains the same, we can write:

Va * (1 + γg * ΔT) = Va

Simplifying, we have:

1 + γg * ΔT = 1

γg * ΔT = 0

Since ΔT cannot be zero (otherwise there would be no change in temperature), we conclude that γg must be zero.

Now, substituting the above information into the equation for V, we have:

V = Vm * (1 + γm * ΔT) + Va

Since γg = 0, we can simplify further:

V = Vm * (1 + γm * ΔT) + Va * (1 + γg * ΔT)

V = Vm * (1 + γm * ΔT) + Va

Since Va remains the same at all temperatures, we can set Va = V - Vm:

V = Vm * (1 + γm * ΔT) + (V - Vm)

Simplifying, we have:

V = Vm + Vm * γm * ΔT + V - Vm

V = V + Vm * γm * ΔT

Subtracting V from both sides, we get:

0 = Vm * γm * ΔT

Since ΔT cannot be zero, we conclude that γm must be zero as well.

Therefore, the volume of the mercury in the flask remains constant at all temperatures.