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Chapter 14: Problem 79

The amount of heat generated during the contraction of muscle in an amphibian's leg is given by $$Q=0.544 \mathrm{mJ}+(1.46 \mathrm{mJ} / \mathrm{cm}) \Delta x$$ where \(\Delta x\) is the length shortened. If a muscle of length \(3.0 \mathrm{cm}\) and mass \(0.10 \mathrm{g}\) is shortened by $1.5 \mathrm{cm}$ during a contraction, what is the temperature rise? Assume that the specific heat of muscle is $4.186 \mathrm{J} /\left(\mathrm{g} \cdot^{\circ} \mathrm{C}\right)$.

Short Answer

Expert verified
Question: Determine the temperature rise during the amphibian muscle's contraction. Answer: The temperature rise during the amphibian muscle's contraction is approximately \(0.00653^{\circ} \mathrm{C}\).

Step by step solution

01

Find the heat generated by muscle contraction

We are given the formula for the amount of heat generated during the contraction: $$Q=0.544 \mathrm{mJ}+(1.46 \mathrm{mJ} / \mathrm{cm}) \Delta x$$ The muscle is shortened by \(\Delta x = 1.5 \mathrm{cm}\). Substituting \(\Delta x\) into the formula, we get: $$Q=0.544 \mathrm{mJ}+(1.46 \mathrm{mJ} / \mathrm{cm}) (1.5 \mathrm{cm})$$
02

Calculate the heat generated

To calculate the heat generated during the muscle contraction, perform the calculations: $$Q=0.544 \mathrm{mJ}+(1.46 \mathrm{mJ} / \mathrm{cm}) (1.5 \mathrm{cm})$$ $$Q=(0.544 + (1.46)(1.5)) \mathrm{mJ}$$ $$Q=(0.544 + 2.19) \mathrm{mJ}$$ $$Q=2.734 \mathrm{mJ}$$ Convert this value to Joules (1 mJ = 0.001 J): $$Q=0.002734 \mathrm{J}$$
03

Use the specific heat to find the temperature rise

We are given the specific heat of muscle, \(C_m = 4.186 \mathrm{J} /\left(\mathrm{g} \cdot^{\circ} \mathrm{C}\right)\), and the mass of the muscle, \(m = 0.10 \mathrm{g}\). The temperature rise, \(\Delta T\), can be found using the formula: $$Q = mC_m\Delta T$$ Rearranging this formula to solve for \(\Delta T\): $$\Delta T=\frac{Q}{mC_m}$$ Now, substitute the values of \(Q, m,\) and \(C_m\) into the formula: $$\Delta T=\frac{0.002734 \mathrm{J}}{0.10 \mathrm{g} \times 4.186 \mathrm{J} /\left(\mathrm{g} \cdot^{\circ} \mathrm{C}\right)}$$
04

Calculate the temperature rise

Finish the calculation to determine the temperature rise during muscle contraction: $$\Delta T=\frac{0.002734 \mathrm{J}}{0.10 \mathrm{g} \times 4.186 \mathrm{J} /\left(\mathrm{g} \cdot^{\circ} \mathrm{C}\right)}$$ $$\Delta T=\frac{0.002734}{0.4186}^{\circ} \mathrm{C}$$ $$\Delta T \approx 0.00653^{\circ} \mathrm{C}$$
05

State the final answer

The temperature rise during the amphibian muscle's contraction is approximately \(0.00653^{\circ} \mathrm{C}\).

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