An aluminum cup of `cm^3` capacity is completely filled with glycerine at `22^@C` . How much glycerine , if any will spill out of the cup if the temperature of both the cup and the glycerine is increased to `28^@C` ? ( The coefficient of volume expansion of glycerine is `5.1xx10^(-4)//""^@C` )

Answer

To solve the problem of how much glycerine will spill out of the aluminum cup when the temperature is increased from 22°C to 28°C, we can follow these steps:

### Step 1: Identify the given values
- Capacity of the aluminum cup (Vc) = 100 cm³
- Initial temperature (T1) = 22°C
- Final temperature (T2) = 28°C
- Coefficient of volume expansion of glycerine (β) = 5.1 × 10^(-4) °C^(-1)
- Coefficient of linear expansion of aluminum (α) = 23 × 10^(-6) °C^(-1)

### Step 2: Calculate the change in temperature (ΔT)
\[ \Delta T = T2 - T1 = 28°C - 22°C = 6°C \]

### Step 3: Calculate the change in volume of glycerine (ΔVg)
Using the formula for volume expansion:
\[ \Delta Vg = \beta \times Vc \times \Delta T \]
Substituting the values:
\[ \Delta Vg = (5.1 \times 10^{-4} \, °C^{-1}) \times (100 \, cm^3) \times (6 \, °C) \]
\[ \Delta Vg = 5.1 \times 10^{-4} \times 100 \times 6 = 0.306 \, cm^3 \]

### Step 4: Calculate the change in volume of the aluminum cup (ΔVc)
The change in volume of the aluminum cup can be calculated using:
\[ \Delta Vc = 3 \alpha \times Vc \times \Delta T \]
Substituting the values:
\[ \Delta Vc = 3 \times (23 \times 10^{-6} \, °C^{-1}) \times (100 \, cm^3) \times (6 \, °C) \]
\[ \Delta Vc = 3 \times 23 \times 10^{-6} \times 100 \times 6 = 0.0414 \, cm^3 \]

### Step 5: Determine the volume of glycerine that spills out
To find the volume of glycerine that spills out, we subtract the change in volume of the cup from the change in volume of glycerine:
\[ \text{Volume spilled} = \Delta Vg - \Delta Vc \]
Substituting the values:
\[ \text{Volume spilled} = 0.306 \, cm^3 - 0.0414 \, cm^3 = 0.2646 \, cm^3 \]

### Conclusion
Approximately 0.265 cm³ of glycerine will spill out of the aluminum cup when the temperature is increased from 22°C to 28°C.

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