a gold nugget and a silver nugget have a combined mass of 14.9g. The two nuggets are heated to 62C and dropped into 15.0mL of H2O at 23.5C. When thermal equilibrium is reached, the temperature of the H2O is 25.0C. Calculate the mass of each nugget. Denisty of H2O=0998g/mL Specific heat: Gold=0.128H.g/C; Silver 0.235J/gC;H2)=4.184J/gC.
mass Au + mass Ag = 14.9; therefore,
mass Au = (14.9-mass Ag) and you will substitute 14.9-mass Ag for mass Au in the below equation.
The general set up is
[mass Au x sp.h. Au x (Tfinal-Tinitial)] + [mass Ag x sp.h. Ag x (Tfinal-Tintial)] + [mass H2O x sp.h. H2O x (Tfinal-Tinitial)] = 0
You have Tfinal and Tinitial, specific heat Au and specific heat Ag and specific heat H2O; the only unknowns are mass Au and mass Ag. For mass Au substitute 14.9-mass Ag which leaves just one unknown, namely mass Ag. Solve for that, and using mass Au + mass Ag = 14.9 you can solve for the Au. The mass H2O is volume x density = ? (Note: when you substitute mass Au = 14.9-mass Ag in that one place, remember that sp.h. Au still goes in that slot)
Post your work if you get stuck.
To calculate the mass of each nugget, we can use the principle of conservation of energy. The heat lost by the nuggets will be equal to the heat gained by the water.
First, let's calculate the heat gained by the water. We can use the formula:
Q = mcΔT
Where:
Q = heat gained by the water (in joules)
m = mass of the water (in grams)
c = specific heat capacity of water (in J/g°C)
ΔT = change in temperature of the water (in °C)
Given:
Mass of water (m) = 15.0 mL = 15.0 g (since the density of water is 0.998 g/mL, we can assume 1 mL is approximately equal to 1 g)
Specific heat capacity of water (c) = 4.184 J/g°C
Change in temperature of water (ΔT) = 25.0°C - 23.5°C = 1.5°C
Plugging in these values, we get:
Q = (15.0 g)(4.184 J/g°C)(1.5°C)
Q = 94.26 J
Next, let's calculate the heat lost by the nuggets. We can use the formula:
Q = mcΔT
For gold:
Specific heat capacity of gold (c) = 0.128 J/g°C
Change in temperature of gold (ΔT) = 62°C - 25.0°C = 37.0°C
Plugging in these values, we get:
Q = mcΔT
Q_gold = m_gold(0.128 J/g°C)(37.0°C)
Similarly, for silver:
Specific heat capacity of silver (c) = 0.235 J/g°C
Change in temperature of silver (ΔT) = 62°C - 25.0°C = 37.0°C
Plugging in these values, we get:
Q = mcΔT
Q_silver = m_silver(0.235 J/g°C)(37.0°C)
Now, since the nuggets have a combined mass of 14.9 g, we can write:
Q_gold + Q_silver = Q
m_gold(0.128 J/g°C)(37.0°C) + m_silver(0.235 J/g°C)(37.0°C) = 94.26 J
We also know that the combined mass of the nuggets is 14.9 g:
m_gold + m_silver = 14.9 g
Now, we can solve this system of equations to find the mass of each nugget. Subtract the equation for m_gold from the equation for m_silver:
m_silver - m_gold = 14.9 g - m_gold
Substitute this into the equation for Q:
(0.235 J/g°C)(37.0°C)(14.9 g - m_gold) + (0.128 J/g°C)(37.0°C)(m_gold) = 94.26 J
Now, solve for m_gold:
(0.235 J/g°C)(37.0°C)(14.9 g - m_gold) + (0.128 J/g°C)(37.0°C)(m_gold) = 94.26 J
Multiply out the terms:
(8.695 J)(14.9 g - m_gold) + (4.736 J)(m_gold) = 94.26 J
Distribute and combine like terms:
129.535 J - 8.695 Jm_gold + 4.736 Jm_gold = 94.26 J
Combine like terms:
-3.959 Jm_gold = -35.275 J
Divide both sides by -3.959 J:
m_gold = 8.89 g
Now, substitute this back into the equation for m_silver:
m_silver = 14.9 g - m_gold
m_silver = 14.9 g - 8.89 g
m_silver = 6.01 g
Therefore, the mass of the gold nugget is approximately 8.89 g, and the mass of the silver nugget is approximately 6.01 g.