Question

A metal rod with a diameter of \(2.30 \mathrm{cm}\) and length of $1.10 \mathrm{m}\( has one end immersed in ice at \)32.0^{\circ} \mathrm{F}$ and the other end in boiling water at \(212^{\circ} \mathrm{F}\). If the ice melts at a rate of 1.32 g every 175 s, what is the thermal conductivity of this metal? Identify the metal. Assume there is no heat lost to the surrounding air.

Short answer

Answer: The thermal conductivity of the metal rod is 0.67 J/(cm s °C), and it is most likely made of iron.

Step-by-step solution

Convert Fahrenheit temperatures to Celsius

First, we need to convert the Fahrenheit temperatures to Celsius. This is because the units of thermal conductivity, which we will be calculating, typically depend on the Celsius scale. To convert from Fahrenheit to Celsius, use the formula: Celsius = (Fahrenheit - 32) * 5/9 For the ice: Celsius_ice = (32 - 32) * 5/9 = 0°C For the boiling water: Celsius_water = (212 - 32) * 5/9 = 100°C

Determine potential energy required to melt ice

Now we will determine the amount of potential energy Q required to melt the ice. Use the formula: Q = mass × specific_heat_of_ice × temperature_difference Where mass is given in the problem statement as 1.32 g and the specific heat of ice is 334 J/g. Potential_energy_to_melt_ice = 1.32 g × 334 J/g = 440.88 J

Determine rate of heat flow

The heat flow rate, which represents the amount of heat transferred per unit of time, can be calculated using the formula below: Heat_flow_rate = Potential_energy_to_melt_ice / time Where time is given as 175 seconds. Heat_flow_rate = 440.88 J / 175 s = 2.52 J/s

Calculate thermal conductivity

Now we will use the formula for heat conduction through a cylindrical rod: Heat_flow_rate = (thermal_conductivity × area × temperature_difference) / length Solving for thermal_conductivity with the information given in the problem: thermal_conductivity = (Heat_flow_rate × length) / (area × temperature_difference) First, we need to calculate the area of the rod. The area of a cylinder can be found using the formula: Area = π × (diameter / 2)^2 Area = π × (2.3 cm / 2)^2 = 4.15 cm² We also need to convert the length to centimeters: Length = 1.1 m × 100 cm/m = 110 cm Now we can plug in all the values we obtained so far: thermal_conductivity = (2.52 J/s × 110 cm) / (4.15 cm² × 100°C) = 0.67 J/(cm s °C)

Identify the metal

Now we can compare the thermal conductivity value we found to known thermal conductivity values of different metals: Aluminum: 2.37 J/(cm s °C) Copper: 3.93 J/(cm s °C) Brass: 1.06 J/(cm s °C) Iron: 0.67 J/(cm s °C) etc. The result we obtained is closest to the value for iron, suggesting that the metal rod is made of iron.