A cylinder with a piston is filled with ideal gas. The gas temperature is held at 450 K. The cylinder is heated by an oven through a square metal rod connected between the oven and the cylinder. The rod has sides d = 2 cm and length L = 2.3 m.
(a) If the oven is held at 850 K, and the heat current conducted through the metal rod is 23 W, find the thermal conductivity of the metal in the rod.
k = ? W/m K
HELP: H = k*A*(T2-T1) / L
HELP: H is the heat current; k is thermal conductivity; T2 and T1 are temperatures at two ends; A and L are cross-sectional area and length of the rod.
To find the thermal conductivity of the metal in the rod, you can use the formula:
H = k * A * (T2 - T1) / L
Where:
H is the heat current
k is the thermal conductivity (what we are trying to find)
A is the cross-sectional area of the rod
T2 and T1 are the temperatures at the two ends of the rod
L is the length of the rod
In this case, we are given:
H = 23 W
T2 = 850 K
T1 = 450 K
A = d^2 (since the rod is square-shaped)
Now, let's substitute the values into the formula and solve for k.
H = k * A * (T2 - T1) / L
23 W = k * (2 cm)^2 * (850 K - 450 K) / 2.3 m
First, convert the length and area into consistent units:
2 cm = 0.02 m (since 1 cm = 0.01 m)
(2 cm)^2 = (0.02 m)^2 = 0.0004 m^2
Now substitute the values:
23 W = k * 0.0004 m^2 * (850 K - 450 K) / 2.3 m
Simplify:
23 W = k * 0.0004 m^2 * 400 K / 2.3 m
23 W = k * 0.0004 m * 400 K / 2.3
23 W = k * 0.4 m * 400 K / 2.3
23 W = k * 0.4 * 400 K / 2.3
23 W = k * 0.4 * (400/2.3) K
23 W = k * 0.4 * 173.913 K
23 W = k * 69.565 K
Now, divide both sides by 69.565 K to solve for k:
23 W / 69.565 K = k
Approximately:
0.33 W/K = k
Therefore, the thermal conductivity of the metal in the rod is approximately 0.33 W/m K.