An aluminum cup is filled to the brim with precisely 0.400 liters of mercury at a temperature of 20 °C. The cup and mercury are then allowed to warm up to a temperature of 50 °C
Calculate the change in volume of the cup and the change in volume of the mercury?
How much mercury over flows the cup?
To calculate the change in volume of the cup and the change in volume of the mercury, you will need to know the coefficients of thermal expansion for both aluminum and mercury.
The coefficient of thermal expansion (α) is a property of a material that measures how much the material expands or contracts when temperature changes. It is usually given in units of per degree Celsius (°C).
The formula to calculate the change in volume is given by:
ΔV = V * α * ΔT,
Where:
ΔV is the change in volume,
V is the initial volume,
α is the coefficient of thermal expansion,
ΔT is the change in temperature.
For aluminum, the coefficient of thermal expansion is approximately 0.000023 (1/°C).
For mercury, the coefficient of thermal expansion is approximately 0.000181 (1/°C).
Now, let's calculate the change in volume of the cup and the change in volume of the mercury.
Change in volume of the cup:
ΔV_cup = V_cup * α_aluminum * ΔT,
where V_cup is the initial volume of the cup, α_aluminum is the coefficient of thermal expansion for aluminum, and ΔT is the change in temperature.
Given:
V_cup = 0.400 liters,
α_aluminum = 0.000023 (1/°C),
ΔT = (50 °C - 20 °C) = 30 °C.
Substituting the values:
ΔV_cup = 0.400 * 0.000023 * 30 = 0.000276 liters.
Therefore, the change in volume of the cup is 0.000276 liters.
Now, let's calculate the change in volume of the mercury.
Change in volume of the mercury:
ΔV_mercury = V_mercury * α_mercury * ΔT,
where V_mercury is the initial volume of the mercury, α_mercury is the coefficient of thermal expansion for mercury, and ΔT is the change in temperature.
Given:
V_mercury = 0.400 liters,
α_mercury = 0.000181 (1/°C),
ΔT = (50 °C - 20 °C) = 30 °C.
Substituting the values:
ΔV_mercury = 0.400 * 0.000181 * 30 = 0.002181 liters.
Therefore, the change in volume of the mercury is 0.002181 liters.
To calculate the amount of mercury that overflows the cup, you need to determine the difference between the change in volume of the mercury and the change in volume of the cup.
Overflow of mercury:
Overflow = ΔV_mercury - ΔV_cup
Overflow = 0.002181 - 0.000276 = 0.001905 liters.
Therefore, approximately 0.001905 liters of mercury overflows the cup.