To solve the problem of calculating the mass of arachidonic acid needed to warm a 500 kg bear from 5°C to 25°C, we will follow these steps:
### Step 1: Calculate the heat required to raise the temperature of the bear.
The formula to calculate the heat (q) required is:
\[ q = m \cdot c \cdot \Delta T \]
where:
- \( m \) = mass of the bear in grams (500 kg = 500,000 g)
- \( c \) = specific heat capacity (4.18 J/g·K)
- \( \Delta T \) = change in temperature (25°C - 5°C = 20°C)
Substituting the values:
\[ q = 500,000 \, \text{g} \cdot 4.18 \, \text{J/g·K} \cdot 20 \, \text{K} \]
\[ q = 500,000 \cdot 4.18 \cdot 20 \]
\[ q = 418,000,000 \, \text{J} \]
\[ q = 418,000 \, \text{kJ} \] (since 1 kJ = 1000 J)
### Step 2: Calculate the heat of combustion of arachidonic acid.
The reaction for the combustion of arachidonic acid is:
\[ C_{20}H_{32}O_{2} + 27 O_{2} \rightarrow 20 CO_{2} + 16 H_{2}O \]
Using the enthalpy of formation values provided:
- \( \Delta H_f^{\circ} \) of \( CO_2(g) = -393.5 \, \text{kJ/mol} \)
- \( \Delta H_f^{\circ} \) of \( H_2O(l) = -285.8 \, \text{kJ/mol} \) (calculated from \( -571.6 \, \text{kJ/mol} \) for 2 moles)
- \( \Delta H_f^{\circ} \) of \( C_{20}H_{32}O_{2} = -636 \, \text{kJ/mol} \)
The heat of combustion \( \Delta H_{comb} \) can be calculated as:
\[ \Delta H_{comb} = [20 \cdot (-393.5) + 16 \cdot (-285.8)] - [-636] \]
Calculating this gives:
\[ \Delta H_{comb} = [(-7870) + (-4572.8) + 636] \]
\[ \Delta H_{comb} = -11806.8 \, \text{kJ/mol} \]
### Step 3: Calculate the number of moles of arachidonic acid required.
Using the heat of combustion, we can find the number of moles (n) of arachidonic acid needed to provide the required heat:
\[ q = n \cdot \Delta H_{comb} \]
Rearranging gives:
\[ n = \frac{q}{|\Delta H_{comb}|} \]
Substituting the values:
\[ n = \frac{418 \, \text{kJ}}{11806.8 \, \text{kJ/mol}} \]
\[ n \approx 0.0354 \, \text{mol} \]
### Step 4: Calculate the mass of arachidonic acid required.
To find the mass (W) of arachidonic acid, we use the formula:
\[ W = n \cdot M \]
where M is the molar mass of arachidonic acid. The molar mass can be calculated as follows:
- C: 20 × 12.01 g/mol = 240.2 g/mol
- H: 32 × 1.008 g/mol = 32.256 g/mol
- O: 2 × 16.00 g/mol = 32.00 g/mol
Thus, the molar mass \( M \) of \( C_{20}H_{32}O_{2} \) is:
\[ M = 240.2 + 32.256 + 32.00 = 304.456 \, \text{g/mol} \]
Now, substituting the values:
\[ W = 0.0354 \, \text{mol} \cdot 304.456 \, \text{g/mol} \]
\[ W \approx 10.77 \, \text{g} \]
### Final Answer:
The mass of arachidonic acid needed to warm the bear is approximately **10.77 grams**.
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