To find the coefficient of volume expansion of mercury, we need to understand how the volume of both the mercury and the glass vessel changes with temperature. The coefficient of volume expansion is a measure of how much a substance expands per degree change in temperature. In this case, we will use the information given about the glass vessel and the overflow of mercury to calculate the coefficient of volume expansion for mercury.
Understanding the Problem
We have a glass vessel with dimensions of 10 cm x 10 cm x 10 cm, which means its initial volume is:
Volume of the glass vessel (V_glass) = 10 cm x 10 cm x 10 cm = 1000 cm³
At 0 degrees Celsius, this vessel is filled with mercury. When the temperature increases to 10 degrees Celsius, 1.6 cc (or cm³) of mercury overflows. This indicates that the volume of mercury has increased due to the temperature rise.
Calculating the Change in Volume of Mercury
The volume of mercury that overflows is equal to the increase in volume of the mercury due to the temperature change. Therefore, we can say:
ΔV_mercury = 1.6 cm³
Volume Expansion of the Glass Vessel
As the temperature increases, the glass vessel also expands, but since the coefficient of linear expansion of glass is given, we can calculate the change in volume of the glass vessel using the formula:
ΔV_glass = V_initial * 3 * α_glass * ΔT
Where:
- V_initial = initial volume of the glass vessel = 1000 cm³
- α_glass = coefficient of linear expansion of glass = 0.0000065 /°C
- ΔT = change in temperature = 10°C - 0°C = 10°C
Now, substituting the values into the formula:
ΔV_glass = 1000 cm³ * 3 * 0.0000065 /°C * 10°C
ΔV_glass = 1000 * 3 * 0.0000065 * 10 = 0.195 cm³
Calculating the Total Volume Change of Mercury
The total volume change of mercury can be expressed as:
ΔV_total = ΔV_mercury - ΔV_glass
Substituting the values we have:
ΔV_total = 1.6 cm³ - 0.195 cm³ = 1.405 cm³
Finding the Coefficient of Volume Expansion of Mercury
The coefficient of volume expansion (β) can be calculated using the formula:
β = ΔV / (V_initial * ΔT)
Where:
- ΔV = total change in volume of mercury = 1.405 cm³
- V_initial = initial volume of mercury = 1000 cm³
- ΔT = change in temperature = 10°C
Now substituting these values into the formula:
β = 1.405 cm³ / (1000 cm³ * 10°C)
β = 1.405 / 10000 = 0.0001405 /°C
Final Result
The coefficient of volume expansion of mercury is approximately 0.0001405 /°C. This means that for every degree Celsius increase in temperature, the volume of mercury expands by about 0.0001405 times its original volume. Understanding these concepts helps us appreciate how materials behave under temperature changes, which is crucial in various scientific and engineering applications.
