Question

a 6.22 kg piece of copper metal is heated from 18.5 °c to 340.3 °c. calculate the heat absorbed (in kj) by the 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.

kj

Explanation:

Step1: Recall the formula for heat absorption

The formula for heat absorbed \( q \) is \( q = mc\Delta T \), where \( m \) is mass, \( c \) is specific heat capacity, and \( \Delta T \) is the change in temperature. For copper, the specific heat capacity \( c = 0.385 \, \text{J/g}^\circ\text{C} \) (from reference tables). First, convert the mass from kg to g: \( m = 6.22 \, \text{kg} = 6220 \, \text{g} \).

Step2: Calculate the change in temperature

\( \Delta T = T_f - T_i = 340.3^\circ\text{C} - 18.5^\circ\text{C} = 321.8^\circ\text{C} \)

Step3: Substitute values into the heat formula

\( q = mc\Delta T = 6220 \, \text{g} \times 0.385 \, \text{J/g}^\circ\text{C} \times 321.8^\circ\text{C} \)
First, calculate \( 6220 \times 0.385 = 2394.7 \)
Then, \( 2394.7 \times 321.8 \approx 770614.46 \, \text{J} \)

Step4: Convert joules to kilojoules

Since \( 1 \, \text{kJ} = 1000 \, \text{J} \), divide by 1000: \( q = \frac{770614.46 \, \text{J}}{1000} = 770.61446 \, \text{kJ} \)
Now, consider significant digits. The mass (6.22 kg) has 3 significant digits, temperature values (18.5, 340.3) have 3 and 4, but the least number of significant digits in multiplication/division is 3 (from mass). So we round to 3 significant digits.
\( 770.61446 \, \text{kJ} \approx 771 \, \text{kJ} \) (Wait, wait, let's check the calculation again. Wait, 6.22 kg is 6220 g (4 significant digits? Wait, 6.22 kg is 6220 g? No, 6.22 kg is 6.22 × 1000 g = 6220 g, but 6.22 has 3 significant digits, so 6220 g (the trailing zero is ambiguous, but since we converted from 6.22 kg, it's 6220 g with 3 significant digits (the 6,2,2). The specific heat is 0.385 (3 sig figs), ΔT is 321.8 (4 sig figs). So when multiplying, the result should have 3 sig figs. Let's recalculate:

\( m = 6.22 \, \text{kg} = 6220 \, \text{g} \) (3 sig figs, since 6.22 has 3)

\( c = 0.385 \, \text{J/g}^\circ\text{C} \) (3 sig figs)

\( \Delta T = 340.3 - 18.5 = 321.8 \, \text{°C} \) (4 sig figs)

\( q = 6220 \times 0.385 \times 321.8 \)

First, 6220 (3 sig figs) × 0.385 (3 sig figs) = 6220 × 0.385 = 2394.7 (but we consider 3 sig figs, so 2390? Wait, no, the rule is that during calculation, we keep extra digits and round at the end. So:

6220 × 0.385 = 6220 × 0.385 = 2394.7

2394.7 × 321.8 = 2394.7 × 321.8

Let's calculate 2394.7 × 321.8:

2394.7 × 300 = 718410

2394.7 × 20 = 47894

2394.7 × 1.8 = 4310.46

Sum: 718410 + 47894 = 766304 + 4310.46 = 770614.46 J

Convert to kJ: 770.61446 kJ

Now, the mass is 6.22 (3 sig figs), specific heat 0.385 (3 sig figs), ΔT 321.8 (4 sig figs). The number of sig figs is determined by the least, which is 3. So 770.61446 rounded to 3 sig figs is 771? Wait, 770.61446, the first three sig figs are 7,7,0? Wait no, 770.61446: the digits are 7 (1st), 7 (2nd), 0 (3rd), 6 (4th)... Wait, no, 770.61446 is 7.7061446 × 10². So to three sig figs, it's 7.71 × 10², which is 771. Wait, but let's check the calculation again. Wait, maybe I made a mistake in the specific heat. Wait, copper's specific heat is 0.385 J/g°C, yes. Mass: 6.22 kg = 6220 g (correct). ΔT: 340.3 - 18.5 = 321.8 (correct). So 6220 * 0.385 = 6220 * 0.385. Let's calculate 6000*0.385=2310, 220*0.385=84.7, so total 2310+84.7=2394.7 (correct). Then 2394.7 * 321.8. Let's do 2394.7 * 321.8:

321.8 * 2000 = 643600

321.8 * 394.7 =? Wait, no, 2394.7 is 2000 + 394.7. So 321.8*2000=643600, 321.8*394.7≈321.8*400=128720, minus 321.8*5.3≈1705.54, so 128720 - 1705.54=127014.46. Then total is 643600 + 127014.46=770614.46 J, which is 770.61446 kJ. Rounding to three significant digits: 771 kJ? Wait, but 6.22 has three, 0.385 has three, 321.8 has four. So the result should have three. 770.61446, the third significant digit is 0? Wait no, 770.61446: the first significant digit is 7, second 7, third 0? Wait, no, trailing zeros in a decimal number are significant, but here it's 770.61446, which is 770.61446, so the digits are 7 (1), 7 (2), 0 (3), 6 (4), etc. Wait, no, 770.61446 is equal to 7.7061446 × 10², so in scientific notation, the first three significant digits are 7,7,0? No, 7.7061446 × 10²: the first digit is 7 (1), second 7 (2), third 0 (3), fourth 6 (4). Wait, that can't be. Wait, no, 770.61446: the number is 7 hundred, 7 ten, 0 one, 6 tenth, etc. So the significant digits are 7,7,0,6,... Wait, but when the number is written as 770.61446, the trailing zero after the 77 is between two non-zero digits? No, 770.61446: the zero is after 77 and before the decimal? Wait, no, 770.61446 is 770 point 61446. So the digits are 7 (hundreds place), 7 (tens place), 0 (ones place), 6 (tenths), etc. So the significant digits: all non-zero digits are significant, and zeros between non-zero digits are significant. So 7,7,0,6,1,4,4,6 are all significant? Wait, no, the zero here is between two non-zero digits (7 and 6), so it is significant. Wait, but the mass is 6.22 (three sig figs), specific heat 0.385 (three), ΔT 321.8 (four). So when multiplying, the number of sig figs is determined by the least, which is three. So we need to round 770.61446 to three sig figs. Let's count: first sig fig 7, second 7, third 0? Wait, no, 770.61446: the first three sig figs are 7,7,0? But 770.61446 rounded to three sig figs would be 771? Wait, no, 770.61446 is closer to 771 than 770? Wait, 770.61446 - 770 = 0.61446, 771 - 770.61446 = 0.38554. So it's closer to 771. So the answer is 771 kJ? Wait, but let's check with exact calculation:

Wait, maybe I made a mistake in the specific heat. Wait, copper's specific heat is 0.385 J/g°C, correct. Mass: 6.22 kg = 6220 g. ΔT: 340.3 - 18.5 = 321.8.

So q = 6220 * 0.385 * 321.8

Calculate 6220 * 0.385 = 6220 * 385 / 1000 = (6220 * 385) / 1000

6220 * 385: 6220 * 300 = 1,866,000; 6220 * 80 = 497,600; 6220 * 5 = 31,100. Sum: 1,866,000 + 497,600 = 2,363,600 + 31,100 = 2,394,700. Then divide by 1000: 2394.7 J/g°C? No, wait, no: 0.385 J/g°C, so 6220 g * 0.385 J/g°C = 6220 * 0.385 J/°C = 2394.7 J/°C. Then multiply by ΔT (321.8 °C): 2394.7 J/°C * 321.8 °C = 2394.7 * 321.8 J.

2394.7 * 321.8:

Let's do 2394.7 * 300 = 718,410

2394.7 * 20 = 47,894

2394.7 * 1.8 = 4,310.46

Sum: 718,410 + 47,894 = 766,304 + 4,310.46 = 770,614.46 J

Convert to kJ: 770,614.46 J / 1000 = 770.61446 kJ

Now, significant digits: mass is 6.22 (3), specific heat 0.385 (3), ΔT 321.8 (4). So the least number of significant digits is 3. So we round 770.61446 to 3 significant digits. The first three significant digits are 7, 7, 0? Wait, no, 770.61446: the digits are 7 (1), 7 (2), 0 (3), 6 (4), etc. Wait, but 770.61446 is 7.7061446 × 10², so in scientific notation, the first three significant digits are 7, 7, 0? No, 7.7061446 × 10²: the first digit is 7 (1), second 7 (2), third 0 (3), fourth 6 (4). Wait, that's not right. Wait, 770.61446: the zero is between two non-zero digits, so it is significant. So the number of significant digits is 8? No, no, the mass is 6.22 (three sig figs), so when we multiply, the result should have three sig figs. So 770.61446 rounded to three sig figs: look at the fourth digit, which is 6, so we round up the third digit. The third digit is 0? Wait, no, 770.61446: digits are 7 (1), 7 (2), 0 (3), 6 (4), 1 (5), 4 (6), 4 (7), 6 (8). Wait, I think I made a mistake here. Wait, 770.61446: the first three significant digits are 7, 7, and 0? But 770.61446 is equal to 770.61446, so the significant digits are 7,7,0,6,1,4,4,6. Wait, no, the zero is after the 77 and before the decimal? No, it's 770.61446, so the zero is in the ones place. So when rounding to three significant digits, we look at the fourth digit (6) to decide. The first three digits are 7,7,0. The fourth digit is 6, which is more than 5, so we round the third digit up. So 0 becomes 1, so 771. So the heat absorbed is approximately 771 kJ. Wait, but let's check with another approach. Maybe the specific heat is in kJ/kg°C? Let's try that. If c = 0.385 kJ/kg°C, then m = 6.22 kg, ΔT = 321.8 °C. Then q = mcΔT = 6.22 kg * 0.385 kJ/kg°C * 321.8 °C. Let's calculate that: 6.22 * 0.385 = 2.3947, then 2.3947 * 321.8 ≈ 770.61446 kJ. Same result. So whether we use g and J or kg and kJ, we get the same value. So rounding to three significant digits, it's 771 kJ. Wait, but 6.22 has three, 0.385 has three, 321.8 has four. So the answer should have three significant digits. So 771 kJ.

Answer:

771