A glass windowpane in a home is 0.62 cm thick and has dimensions of 1.30 m × 1.72 m. On a certain day, the indoor temperature is 26°C and the outdoor temperature is 0°C. (Assume the thermal conductivity of the glass is 0.8 J/s · m · °C.)

(a) What is the rate at which energy is transferred by heat through the glass?
(b) How much energy is lost through the window in one day, assuming the temperatures inside and outside remain constant?

(a) For the rate, use the formula

dQ/dt = k A dT/ex

(b) For total energy loss, multiply by time (in seconds)

To find the rate at which energy is transferred by heat through the glass (Q_dot), we can use the formula:

Q_dot = (k * A * ΔT) / L

Where:
- Q_dot is the rate of energy transfer (in watts, or J/s).
- k is the thermal conductivity of the glass (in J/s · m · °C).
- A is the cross-sectional area of the window (in m²).
- ΔT is the temperature difference between the indoor and outdoor temperatures (in °C).
- L is the thickness of the glass (in meters).

Let's plug in the values from the problem and solve for Q_dot:

(a) Rate of energy transfer:
Q_dot = (0.8 J/s · m · °C) * (1.30 m * 1.72 m) * (26°C - 0°C) / 0.62 cm

First, we need to convert the thickness of the glass to meters:
0.62 cm = 0.62 cm * (1 m / 100 cm) = 0.0062 m

Now we can substitute the values into the formula:

Q_dot = (0.8 J/s · m · °C) * (1.30 m * 1.72 m) * (26°C - 0°C) / 0.0062 m

Calculating this will give us the rate at which energy is transferred by heat through the glass (Q_dot) in watts (or J/s).

To find the energy lost through the window in one day, we can multiply Q_dot by the number of seconds in a day:

(b) Energy lost in one day:
Energy_lost = Q_dot * 24 hours * 60 minutes * 60 seconds

Let's calculate the energy lost through the window in one day using the value we found for Q_dot in part (a).

To solve this problem, we can use the formula for heat transfer through a material:

Q = (k * A * ΔT) / L

Where:
Q is the rate of heat transfer in watts (W)
k is the thermal conductivity of the glass in joules per second, per meter, per degree Celsius (J/s · m · °C)
A is the area of the glass in square meters (m²)
ΔT is the temperature difference in degrees Celsius (°C)
L is the thickness of the glass in meters (m)

Let's calculate the answers step-by-step:

(a) What is the rate at which energy is transferred by heat through the glass?

Given:
k = 0.8 J/s · m · °C
A = 1.30 m × 1.72 m = 2.236 m² (rounded to 3 decimal places)
ΔT = (26°C - 0°C) = 26°C
L = 0.62 cm = 0.0062 m

Substituting these values into the formula, we get:

Q = (0.8 J/s · m · °C) * (2.236 m²) * (26°C) / (0.0062 m)
Q ≈ 1835.90 W

Therefore, the rate at which energy is transferred by heat through the glass is approximately 1835.90 watts.

(b) How much energy is lost through the window in one day, assuming the temperatures inside and outside remain constant?

In order to calculate the energy lost through the window over one day, we need to know the duration of one day. Let's assume a typical day is 24 hours.

To calculate the total energy lost in one day, we use the following formula:

Energy_lost = Q * t

Where:
Energy_lost is the total energy lost in joules (J)
Q is the rate of heat transfer in watts (W)
t is the time in seconds (s)

Given:
Q = 1835.90 W
t = 24 hours = 24 * 60 * 60 seconds = 86400 s

Substituting these values into the formula, we get:

Energy_lost = (1835.90 W) * (86400 s)
Energy_lost ≈ 158678560 J

Therefore, assuming the temperatures inside and outside remain constant, the amount of energy lost through the window in one day is approximately 158,678,560 joules.