The volume of a glass vessel is 1000 cc at 20C. What volume of mercury should be poured into it at this temperature so that the volume of the remaing space does not change with temperature ? coefficients of cubical expansion of mercury and glass are 1.8 * 10 ^ -4 / c and 9.0 * 10 ^ -6 / c respectively.

To solve the problem of how much mercury to pour into a glass vessel so that the volume of the remaining space does not change with temperature, we need to consider the concept of thermal expansion. Both the glass and mercury will expand when the temperature increases, but they do so at different rates due to their respective coefficients of cubical expansion.

Understanding Thermal Expansion

Thermal expansion refers to the way materials change in volume in response to temperature changes. The coefficients of cubical expansion for a material indicate how much its volume will change per degree change in temperature. In this case, we have:

• Coefficient of cubical expansion of mercury (β_Hg): 1.8 × 10^-4 /°C
• Coefficient of cubical expansion of glass (β_glass): 9.0 × 10^-6 /°C

Setting Up the Equation

Let’s denote the volume of mercury poured into the glass vessel as V_Hg. The total volume of the glass vessel is 1000 cc. When the temperature changes, both the mercury and the glass will expand. We want to find V_Hg such that the volume of the remaining space (the volume of the glass vessel minus the volume of mercury) does not change with temperature.

At a temperature change of ΔT, the change in volume for mercury and glass can be expressed as:

• Change in volume of mercury: ΔV_Hg = V_Hg × β_Hg × ΔT
• Change in volume of glass: ΔV_glass = V_glass × β_glass × ΔT

Here, V_glass is the volume of the glass vessel, which is 1000 cc, minus V_Hg (the volume of mercury). Thus, we can express the change in volume of glass as:

ΔV_glass = (1000 - V_Hg) × β_glass × ΔT

Equating the Changes in Volume

For the volume of the remaining space to remain constant with temperature, the change in volume of mercury must equal the change in volume of glass:

V_Hg × β_Hg × ΔT = (1000 - V_Hg) × β_glass × ΔT

We can simplify this equation by canceling out ΔT (assuming it is not zero):

V_Hg × β_Hg = (1000 - V_Hg) × β_glass

Solving for V_Hg

Now, let’s plug in the values for the coefficients:

V_Hg × (1.8 × 10^-4) = (1000 - V_Hg) × (9.0 × 10^-6)

Expanding this gives us:

1.8 × 10^-4 V_Hg = 9000 - 9.0 × 10^-6 V_Hg

Now, combining like terms:

1.8 × 10^-4 V_Hg + 9.0 × 10^-6 V_Hg = 9000

(1.8 × 10^-4 + 9.0 × 10^-6) V_Hg = 9000

Calculating the left side:

1.8 × 10^-4 = 0.00018

9.0 × 10^-6 = 0.000009

So, 0.00018 + 0.000009 = 0.000189

Now we can solve for V_Hg:

0.000189 V_Hg = 9000

V_Hg = 9000 / 0.000189

V_Hg ≈ 47670.37 cc

Final Consideration

However, this volume exceeds the capacity of the glass vessel, indicating that the assumption of constant volume in the remaining space cannot be met with the given coefficients of expansion. Therefore, it is impossible to maintain the volume of the remaining space unchanged with temperature using mercury in this glass vessel.

In summary, while we can calculate the theoretical volume of mercury needed, the physical constraints of the glass vessel limit the practical application of this scenario. Understanding thermal expansion helps us appreciate the complexities involved in real-world applications.