Answer
We can use the principle of conservation of energy to solve this problem. The energy lost by the copper is equal to the energy gained by the zinc. The formula for the energy transferred is:
Q = mcpΔT
Where Q is the amount of heat transferred, m is the mass of the object, cp is the specific heat capacity, and ΔT is the change in temperature.
Let's first calculate the energy transferred from the copper to the zinc:
Q1 = m1 * cp1 * ΔT1
Q1 = 755 kg * 572 J/kg·K * (1350°C - 1170°C)
Where cp1 is the specific heat capacity of copper.
The energy gained by the zinc is:
Q2 = m2 * cp2 * ΔT2
Q2 = m2 * 497 J/kg·K * (1170°C - 469°C)
Where cp2 is the specific heat capacity of zinc.
Since there is no heat loss, the energy gained by the zinc is equal to the energy lost by the copper:
Q1 = Q2
755 kg * 572 J/kg·K * (1350°C - 1170°C) = m2 * 497 J/kg·K * (1170°C - 469°C)
Simplifying the equation, we get:
755 * 572 * 180 = m2 * 497 * 701
Solving for m2, we get:
m2 = 962 kg
Therefore, the mass of zinc added is 962 kg. The total mass of the alloy is:
m_total = m_copper + m_zinc
m_total = 755 + 962
m_total = 1717 kg
The percent of the alloy's mass that is zinc is:
percent_zinc = m_zinc / m_total * 100
percent_zinc = 962 / 1717 * 100
percent_zinc = 56.0%
Therefore, approximately 56.0% of the alloy's mass is zinc.
Work Step by Step
We can use the principle of conservation of energy to solve this problem. The energy lost by the copper is equal to the energy gained by the zinc. The formula for the energy transferred is:
Q = mcpΔT
Where Q is the amount of heat transferred, m is the mass of the object, cp is the specific heat capacity, and ΔT is the change in temperature.
Let's first calculate the energy transferred from the copper to the zinc:
Q1 = m1 * cp1 * ΔT1
Q1 = 755 kg * 572 J/kg·K * (1350°C - 1170°C)
Where cp1 is the specific heat capacity of copper.
The energy gained by the zinc is:
Q2 = m2 * cp2 * ΔT2
Q2 = m2 * 497 J/kg·K * (1170°C - 469°C)
Where cp2 is the specific heat capacity of zinc.
Since there is no heat loss, the energy gained by the zinc is equal to the energy lost by the copper:
Q1 = Q2
755 kg * 572 J/kg·K * (1350°C - 1170°C) = m2 * 497 J/kg·K * (1170°C - 469°C)
Simplifying the equation, we get:
755 * 572 * 180 = m2 * 497 * 701
Solving for m2, we get:
m2 = 962 kg
Therefore, the mass of zinc added is 962 kg. The total mass of the alloy is:
m_total = m_copper + m_zinc
m_total = 755 + 962
m_total = 1717 kg
The percent of the alloy's mass that is zinc is:
percent_zinc = m_zinc / m_total * 100
percent_zinc = 962 / 1717 * 100
percent_zinc = 56.0%
Therefore, approximately 56.0% of the alloy's mass is zinc.
