Question

a 20.3 g sample of an unknown metal and a 28.5 g sample of copper, both at 80.6°c, are added to 102. g of water at 11.2°c in a constant - pressure calorimeter of negligible heat capacity. if the final temperature of the metals and water is 16.9°c, determine the specific heat of the unknown metal. be sure your answer has the correct number of significant digits. note: reference the phase change properties of pure substances table for additional information.

Explanation:

Step1: Recall heat transfer formula

The heat lost by the metals (unknown metal and copper) is equal to the heat gained by the water, since the calorimeter has negligible heat capacity. The formula for heat transfer is \( q = mc\Delta T \), where \( q \) is heat, \( m \) is mass, \( c \) is specific heat, and \( \Delta T \) is the change in temperature.

Let \( c_{unknown} \) be the specific heat of the unknown metal, \( c_{Cu} = 0.385 \, \frac{J}{g\cdot^\circ C} \) (from reference tables), \( m_{unknown} = 20.3 \, g \), \( m_{Cu} = 28.5 \, g \), \( m_{water} = 102. \, g \), \( c_{water} = 4.184 \, \frac{J}{g\cdot^\circ C} \), initial temperature of metals \( T_{metal, initial} = 80.6^\circ C \), initial temperature of water \( T_{water, initial} = 11.2^\circ C \), final temperature \( T_{final} = 16.9^\circ C \).

Heat lost by metals: \( q_{lost} = m_{unknown}c_{unknown}(T_{metal, initial} - T_{final}) + m_{Cu}c_{Cu}(T_{metal, initial} - T_{final}) \)

Heat gained by water: \( q_{gained} = m_{water}c_{water}(T_{final} - T_{water, initial}) \)

Set \( q_{lost} = q_{gained} \):

\( m_{unknown}c_{unknown}(T_{metal, initial} - T_{final}) + m_{Cu}c_{Cu}(T_{metal, initial} - T_{final}) = m_{water}c_{water}(T_{final} - T_{water, initial}) \)

Step2: Calculate temperature changes

For metals: \( \Delta T_{metal} = 80.6 - 16.9 = 63.7^\circ C \)

For water: \( \Delta T_{water} = 16.9 - 11.2 = 5.7^\circ C \)

Step3: Substitute values and solve for \( c_{unknown} \)

First, calculate the heat gained by water:

\( q_{gained} = 102. \, g \times 4.184 \, \frac{J}{g\cdot^\circ C} \times 5.7^\circ C \)

\( q_{gained} = 102. \times 4.184 \times 5.7 \)

\( 102. \times 4.184 = 426.768 \)

\( 426.768 \times 5.7 = 2432.5776 \, J \)

Now, calculate the heat lost by copper:

\( q_{Cu} = 28.5 \, g \times 0.385 \, \frac{J}{g\cdot^\circ C} \times 63.7^\circ C \)

\( 28.5 \times 0.385 = 10.9725 \)

\( 10.9725 \times 63.7 = 699.94825 \, J \)

Let \( q_{unknown} = m_{unknown}c_{unknown}\Delta T_{metal} = 20.3 \times c_{unknown} \times 63.7 \)

Then, \( q_{unknown} + q_{Cu} = q_{gained} \)

\( 20.3 \times c_{unknown} \times 63.7 + 699.94825 = 2432.5776 \)

Subtract \( 699.94825 \) from both sides:

\( 20.3 \times c_{unknown} \times 63.7 = 2432.5776 - 699.94825 = 1732.62935 \)

Calculate \( 20.3 \times 63.7 = 1293.11 \)

Then, \( c_{unknown} = \frac{1732.62935}{1293.11} \approx 1.34 \, \frac{J}{g\cdot^\circ C} \) (wait, let's check calculations again for accuracy)

Wait, recalculating \( q_{gained} \):

\( m_{water}c_{water}\Delta T_{water} = 102. \times 4.184 \times 5.7 \)

102. * 4.184 = 426.768; 426.768 * 5.7: 426.768*5 = 2133.84, 426.768*0.7=298.7376; total = 2133.84 + 298.7376 = 2432.5776 J (correct)

\( q_{Cu} = 28.5 * 0.385 * 63.7 \)

28.5 * 0.385 = 10.9725; 10.9725 * 63.7: 10 * 63.7 = 637, 0.9725 * 63.7 ≈ 62.05825; total ≈ 637 + 62.05825 = 699.05825 J (close to before)

Then \( q_{unknown} = q_{gained} - q_{Cu} = 2432.5776 - 699.05825 = 1733.51935 J \)

\( q_{unknown} = m_{unknown}c_{unknown}\Delta T_{metal} \)

So \( c_{unknown} = \frac{q_{unknown}}{m_{unknown}\Delta T_{metal}} = \frac{1733.51935}{20.3 * 63.7} \)

20.3 * 63.7 = 20.3 * 60 + 20.3 * 3.7 = 1218 + 75.11 = 1293.11

1733.51935 / 1293.11 ≈ 1.341, but wait, maybe I used the wrong specific heat for copper? Wait, copper's specific heat is 0.385 J/g°C, that's correct. Wait, but let's check the calculation again.

Wait, maybe I made a mistake in the sign, but the formula is correct (heat lost by metals = heat gained by water). Let's re-express the equation:

\( (m_{unknown}c_{unknown} + m_{Cu}c_{Cu})\Delta T_{metal} = m_{water}c_{water}\Delta T_{water} \)

So,

\( c_{unknown} = \frac{m_{water}c_{water}\Delta T_{water} - m_{Cu}c_{Cu}\Delta T_{metal}}{m_{unknown}\Delta T_{metal}} \)

Plugging in the numbers:

Numerator: (102. * 4.184 * 5.7) - (28.5 * 0.385 * 63.7)

We have 2432.5776 - 699.05825 = 1733.51935

Denominator: 20.3 * 63.7 = 1293.11

1733.51935 / 1293.11 ≈ 1.341, which rounds to 1.34? Wait, but let's check significant digits. The given values: 20.3 (3 sig figs), 28.5 (3), 102. (3), 80.6 (3), 11.2 (3), 16.9 (3). So the answer should have 3 sig figs. Wait, but maybe my calculation is wrong. Wait, let's use more precise calculation.

Wait, 102. * 4.184 = 426.768; 426.768 * 5.7:

426.768 * 5.7 = 426.768 * (5 + 0.7) = 426.768*5 + 426.768*0.7 = 2133.84 + 298.7376 = 2432.5776 J

28.5 * 0.385 = 10.9725; 10.9725 * 63.7:

10.9725 * 63.7:

First, 10 * 63.7 = 637

0.9725 * 63.7: 0.9 * 63.7 = 57.33; 0.0725 * 63.7 ≈ 4.61825; total ≈ 57.33 + 4.61825 = 61.94825

So total q_Cu = 637 + 61.94825 = 698.94825 J

Then q_unknown = 2432.5776 - 698.94825 = 1733.62935 J

m_unknown * delta T_metal = 20.3 * (80.6 - 16.9) = 20.3 * 63.7 = 20.3 * 60 + 20.3 * 3.7 = 1218 + 75.11 = 1293.11 g·°C

c_unknown = 1733.62935 / 1293.11 ≈ 1.3407 ≈ 1.34 J/g·°C? Wait, but maybe I messed up the specific heat of copper? Wait, no, copper's specific heat is 0.385 J/g°C. Wait, but let's check with another approach.

Wait, maybe the error is in the specific heat of copper? Wait, no, 0.385 is correct. Wait, let's recalculate the numerator and denominator:

Numerator: 2432.5776 - (28.5 * 0.385 * 63.7)

28.5 * 0.385 = 10.9725; 10.9725 * 63.7 = 10.9725 * 63.7

Let's do 10.9725 * 63.7:

63.7 * 10 = 637

63.7 * 0.9725 = 63.7 * (1 - 0.0275) = 63.7 - 63.7*0.0275 = 63.7 - 1.75175 = 61.94825

So total is 637 + 61.94825 = 698.94825

2432.5776 - 698.94825 = 1733.62935

Denominator: 20.3 * 63.7 = 20.3 * 63.7

20 * 63.7 = 1274

0.3 * 63.7 = 19.11

Total = 1274 + 19.11 = 1293.11

1733.62935 / 1293.11 ≈ 1.3407, which is approximately 1.34 J/g·°C. Wait, but I think I made a mistake earlier, maybe the specific heat of copper is 0.387? Wait, no, standard value is 0.385 or 0.387. Let's use 0.387 for copper.

Let's recalculate with c_Cu = 0.387 J/g°C

q_Cu = 28.5 * 0.387 * 63.7

28.5 * 0.387 = 11.0295

11.0295 * 63.7 = 11 * 63.7 + 0.0295 * 63.7 = 700.7 + 1.87915 = 702.57915

Then q_unknown = 2432.5776 - 702.57915 = 1729.99845

c_unknown = 1729.99845 / 1293.11 ≈ 1.338, which is still ~1.34. Wait, but maybe the correct answer is 1.34? Wait, no, let's check an example. Wait, maybe I messed up the formula. The heat lost by the metals is equal to the heat gained by the water, so:

q_metals = -q_water (but since we're equating magnitudes, it's q_metals = q_water)

So,

m_unknown * c_unknown * (T_initial_metal - T_final) + m_Cu * c_Cu * (T_initial_metal - T_final) = m_water * c_water * (T_final - T_initial_water)

Yes, that's correct.

So, solving for c_unknown:

c_unknown = [m_water * c_water * (T_final - T_initial_water) - m_Cu * c_Cu * (T_initial_metal - T_final)] / [m_unknown * (T_initial_metal - T_final)]

Plugging in:

m_water = 102. g, c_water = 4.184 J/g°C, T_final - T_initial_water = 16.9 - 11.2 = 5.7 °C

m_Cu = 28.5 g, c_Cu = 0.385 J/g°C, T_initial_metal - T_final = 80.6 - 16.9 = 63.7 °C

m_unknown = 20.3 g, T_initial_metal - T_final = 63.7 °C

So,

Numerator: (102. * 4.184 * 5.7) - (28.5 * 0.385 * 63.7) = 2432.5776 - 699.05825 = 1733.51935

Denominator: 20.3 * 63.7 = 1293.11

1733.51935 / 1293.11 ≈ 1.34 J/g°C

Wait, but maybe the answer is 1.34? Let me check with a calculator.

1733.51935 ÷ 1293.11 ≈ 1.3407, which is approximately 1.34 when rounded to three significant figures.

Answer:

\boxed{1.34}