To find the coefficient of volume expansion of mercury, we need to analyze the situation step by step. We have a vessel filled with water and mercury, and when heat is added, some water overflows. Let's break down the problem using the information provided.
Understanding the Problem
We start with a vessel containing:
- 500 g of water
- 1000 g of mercury
When 21200 cal of heat is added, 3.52 g of water overflows. Our goal is to calculate the coefficient of volume expansion of mercury.
Key Concepts
1. **Volume Expansion**: The volume expansion of a substance can be expressed as:
ΔV = V₀ * β * ΔT
where ΔV is the change in volume, V₀ is the initial volume, β is the coefficient of volume expansion, and ΔT is the change in temperature.
2. **Heat Transfer**: The heat gained by the water and mercury can be calculated using:
Q = m * c * ΔT
where Q is the heat added, m is the mass, c is the specific heat, and ΔT is the change in temperature.
Calculating the Change in Temperature
First, we need to determine how much heat was used to cause the overflow of water. The mass of water that overflowed is 3.52 g. The specific heat of water is approximately 1 cal/g°C. Therefore, the heat absorbed by the overflowing water can be calculated as:
Q_water = m_water * c_water * ΔT
Rearranging gives us:
ΔT = Q_water / (m_water * c_water)
Substituting the values:
Q_water = 3.52 g * 1 cal/g°C * ΔT
Now, we know that the total heat added is 21200 cal. The heat used to raise the temperature of both water and mercury can be expressed as:
Q_total = Q_water + Q_mercury
Since we are interested in the change in temperature, we can express the heat for mercury as:
Q_mercury = m_mercury * c_mercury * ΔT
Substituting the known values:
Q_mercury = 1000 g * 0.03 cal/g°C * ΔT
Setting Up the Equation
Now we can set up the equation:
21200 = 3.52 * ΔT + 1000 * 0.03 * ΔT
Combining terms gives:
21200 = (3.52 + 30) * ΔT
21200 = 33.52 * ΔT
Solving for ΔT:
ΔT = 21200 / 33.52 ≈ 632.5 °C
Calculating the Change in Volume of Mercury
Next, we need to find the change in volume of mercury. The initial volume of mercury can be calculated using its density:
V_mercury = m_mercury / density_mercury
Substituting the values:
V_mercury = 1000 g / 13.6 g/cm³ ≈ 73.53 cm³
Now we can calculate the change in volume:
ΔV_mercury = V_mercury * β_mercury * ΔT
We need to express β_mercury in terms of ΔV and the other known quantities. Since we don't know β_mercury yet, we can rearrange this equation:
β_mercury = ΔV_mercury / (V_mercury * ΔT)
Finding the Volume Change
The volume change of mercury must equal the volume of water that overflowed. The volume of the overflowed water can be calculated as:
V_water_overflow = m_water_overflow / density_water
Assuming the density of water is approximately 1 g/cm³:
V_water_overflow = 3.52 g / 1 g/cm³ = 3.52 cm³
Final Calculation
Now we can substitute everything back into our equation for β_mercury:
β_mercury = 3.52 cm³ / (73.53 cm³ * 632.5 °C)
Calculating this gives:
β_mercury ≈ 0.0000768 °C⁻¹
Thus, the coefficient of volume expansion of mercury is approximately 7.68 x 10^-5 °C⁻¹.
