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To solve this problem, we need to calculate the amount of energy required to heat the water, then determine how many butane canisters are needed to provide that energy.

### Steps:

1. **Determine the energy needed to heat the water:**
   - Mass of water: 4.0 kg (or 4000 g)
   - Specific heat of water: 4.18 J/g°C
   - Temperature change: The water starts at 25°C (air temperature) and is heated to 100°C (boiling point), so the change in temperature is \( \Delta T = 100°C - 25°C = 75°C \).
   
   The energy required to heat the water can be found using the formula:
   \[
   Q = mc\Delta T
   \]
   Where:
   - \( Q \) is the heat energy (in joules),
   - \( m \) is the mass of water (4000 g),
   - \( c \) is the specific heat capacity (4.18 J/g°C),
   - \( \Delta T \) is the change in temperature (75°C).

   So, the energy required per day is:
   \[
   Q = 4000 \times 4.18 \times 75 = 1,254,000 \text{ J} = 1,254 \text{ kJ}
   \]

2. **Calculate the total energy needed for the 7 days:**
   The energy required for the whole trip is:
   \[
   Q_{\text{total}} = 1,254 \text{ kJ/day} \times 7 \text{ days} = 8,778 \text{ kJ}
   \]

3. **Determine the energy provided by the butane:**
   - Molar mass of butane \( C_4H_{10} \): \( 4 \times 12.01 + 10 \times 1.008 = 58.14 \, \text{g/mol} \).
   - Standard heat of formation (combustion) of butane: \( -125.7 \, \text{kJ/mol} \).

   One canister contains 75 g of butane, so the number of moles of butane per canister is:
   \[
   \frac{75 \, \text{g}}{58.14 \, \text{g/mol}} = 1.29 \, \text{mol}
   \]
   The energy released by burning 1 canister of butane is:
   \[
   1.29 \, \text{mol} \times 125.7 \, \text{kJ/mol} = 162.15 \, \text{kJ}
   \]

4. **Calculate the number of canisters needed:**
   To meet the total energy demand of 8,778 kJ, the number of canisters required is:
   \[
   \frac{8,778 \, \text{kJ}}{162.15 \, \text{kJ/canister}} = 54.14 \, \text{canisters}
   \]

   Since you can't use a fraction of a canister, you will need to round up to the next whole number, which is 55 canisters.

### Final Answer:
The minimum number of fuel canisters you must bring is **55**.

Do you need further details or have any questions?

Here are some related questions you might explore:
1. How would the number of canisters change if the temperature dropped below 25°C?
2. What if the water was starting at 0°C instead of 25°C?
3. How much energy would you save by using a more efficient fuel than butane?
4. How would you calculate the number of canisters if each canister contained a different fuel with a higher heat of formation?
5. What if the specific heat of water changed due to altitude?

**Tip:** Always make sure to account for all variables in real-world applications, such as heat losses due to environmental factors.

On a 7 day wilderness expedition you'll need to heat 4.0 kg of water to the boiling point each day. The air temperature will average 25 °C. You have available canisters of compressed butane (C4H10) fuel, which you'll burn to heat the water. Each canister has 75 g of butane in it. What is the minimum number of fuel canisters you must bring?

Determine how many butane canisters are needed to heat water to boiling on a 7-day expedition. This thermodynamics problem covers key concepts such as energy conversion and stoichiometry. Learn how to apply formulas like Q = mcΔT and calculate moles to ensure you bring enough fuel for the entire trip.

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