Question

calculate the energy required to heat 514.0 g of cyclohexane from 13.1°c to 25.0°c. assume the specific heat capacity of cyclohexane under these conditions is 1.85 j·g⁻¹·k⁻¹. be sure your answer has the correct number of significant digits

Explanation:

Step1: Recall the formula for heat energy

The formula to calculate the heat energy \( q \) required to raise the temperature of a substance is \( q = mc\Delta T \), where \( m \) is the mass, \( c \) is the specific heat capacity, and \( \Delta T \) is the change in temperature. Since the change in Celsius is equal to the change in Kelvin (\( \Delta T (\text{K})=\Delta T (\text{°C}) \)), we can use the Celsius values for \( \Delta T \).

Step2: Calculate the change in temperature

\( \Delta T = T_2 - T_1 = 25.0\ \text{°C} - 13.1\ \text{°C} = 11.9\ \text{°C} \) (which is also \( 11.9\ \text{K} \)).

Step3: Substitute the values into the formula

Given \( m = 514.0\ \text{g} \), \( c = 1.85\ \text{J g}^{-1}\text{K}^{-1} \), and \( \Delta T = 11.9\ \text{K} \).
\[ q = 514.0\ \text{g} \times 1.85\ \text{J g}^{-1}\text{K}^{-1} \times 11.9\ \text{K} \]
First, multiply \( 514.0 \times 1.85 \): \( 514.0 \times 1.85 = 950.9 \).
Then, multiply the result by \( 11.9 \): \( 950.9 \times 11.9 = 11315.71 \).

Step4: Consider significant digits

The mass \( 514.0 \) has 4 significant digits, specific heat \( 1.85 \) has 3, and \( \Delta T = 11.9 \) has 3. The least number of significant digits in the multiplication is 3, so we round the answer to 3 significant digits.
\( 11315.71 \approx 1.13 \times 10^4 \) (or 11300 when rounded to three significant digits? Wait, let's check the multiplication again. Wait, \( 514.0 \times 1.85 = 514 \times 1.85 = (500 + 14) \times 1.85 = 500 \times 1.85 + 14 \times 1.85 = 925 + 25.9 = 950.9 \). Then \( 950.9 \times 11.9 \): let's calculate it more accurately. \( 950.9 \times 12 = 11410.8 \), minus \( 950.9 \times 0.1 = 95.09 \), so \( 11410.8 - 95.09 = 11315.71 \). Now, the specific heat has 3 sig figs, \( \Delta T \) has 3, mass has 4. So the answer should have 3 sig figs. \( 11315.71 \) rounded to 3 sig figs is \( 1.13 \times 10^4 \) J or 11300 J? Wait, 11315.71: the first three significant digits are 1, 1, 3, the next digit is 1, which is less than 5, so we keep it 11300? Wait, no, 11315.71 is 1.131571×10⁴, so with three significant digits, it's 1.13×10⁴ J (or 11300 J when written in standard form, but in scientific notation, 1.13×10⁴). Wait, but let's check the multiplication again. Wait, maybe I made a mistake in the number of significant digits. The mass is 514.0 (four sig figs), specific heat 1.85 (three), ΔT 11.9 (three). When multiplying, the result should have three sig figs. So 514.0 × 1.85 × 11.9. Let's do the calculation with more precision:

514.0 * 1.85 = 514.0 * 1.85 = 950.9 (exact here because 514.0 has a decimal, so 4 sig figs, 1.85 has 3, so 950.9 is 4 sig figs, but when multiplied by 11.9 (3 sig figs), the result should have 3 sig figs.

950.9 * 11.9:

950.9 * 11.9 = 950.9*(12 - 0.1) = 950.9*12 - 950.9*0.1 = 11410.8 - 95.09 = 11315.71

Now, 11315.71 rounded to three significant digits: the first three digits are 1, 1, 3. The next digit is 1, which is less than 5, so we round down. So 11300 J? Wait, no, 1.13×10⁴ J is three significant digits (1,1,3). Alternatively, 11300 J with the trailing zero as a placeholder, but the significant digits are 1,1,3. Wait, maybe the correct calculation is:

514.0 * 1.85 = 950.9

950.9 * 11.9 = let's use a calculator: 950.9 * 11.9 = 950.9 * 11 + 950.9 * 0.9 = 10459.9 + 855.81 = 11315.71

Now, considering significant digits: the least number of significant digits in the factors is 3 (from 1.85 and 11.9), so the answer should have 3 significant digits. So 11315.71 ≈ 1.13 × 10⁴ J (or 11300 J). Wait, but let's check the original numbers:

Mass: 514.0 (4 sig figs)

Specific heat: 1.85 (3 sig figs)

ΔT: 25.0 - 13.1 = 11.9 (3 sig figs, since 25.0 has 3 decimal? No, 25.0 has three significant digits? Wait, 25.0 has three significant digits? No, 25.0 has three? Wait, 25.0: the zero is significant, so it's three? Wait, 25.0 is three significant digits? No, 25.0 has three? Wait, 25.0: the digits are 2,5,0, so three significant digits? Wait, no, 25.0 has three significant digits? Wait, 514.0 has four (the zero after the decimal is significant), 1.85 has three, 25.0 has three, 13.1 has three. So ΔT is 25.0 -13.1 = 11.9 (three significant digits). So when multiplying, the number of significant digits is determined by the least number, which is three (from 1.85 and 11.9). So the answer should have three significant digits. So 11315.71 rounded to three significant digits is 11300? Wait, 11315.71: the first three digits are 1,1,3, the next digit is 1, which is less than 5, so we keep it 11300. Wait, but 1.13 × 10⁴ is also three significant digits. Either way, the value is approximately 1.13 × 10⁴ J or 11300 J.

Answer:

\boxed{1.13 \times 10^4} (or \boxed{11300} depending on the required format, but in scientific notation with three significant digits, \( 1.13 \times 10^4 \) J is appropriate)