Lake Erie contains roughly 4.00 1011 m3 of water.
(a) How much energy is required to raise the temperature of that volume of water from 9.0°C to 11.0°C?
J
(b) How many years would it take to supply this amount of energy by using the 1250-MW exhaust energy of an electric power plant?
yr
For water, density is ρ =1000 kg/m ³,
V =4•10¹¹ m³ => m=ρ•V=1000•4•10¹¹ =4•10¹⁴ kg,
c =4183 J/kg•K, ΔT=11-9 =2 K.
E=Q= m•c• (T₁-T)= m•c• ΔT =4•10¹¹•4183•2=3.3464•10¹⁸ s
(b) E=P•t
t=E/P=3.3464•10¹⁸/1250•10⁶=2.677•10⁹ s=2.677•10⁹/365•24•3600=84.9 years.
(a) Well, let's dive right in! To calculate the energy required to raise the temperature of Lake Erie's water from 9.0°C to 11.0°C, we can use the formula:
Energy = mass × specific heat capacity × change in temperature
Now, the mass of water is given by the volume and density, which is roughly 1.00 g/cm³. Converting the volume to cubic meters, we have:
Mass = volume × density
So, Mass = 4.00 × 10^11 m³ × 1.00 g/cm³
But wait, we need to convert grams to kilograms to keep our calculations swimmingly accurate! That's easy: 1 g = 0.001 kg.
So, Mass = 4.00 × 10^11 m³ × 1.00 g/cm³ × 0.001 kg/g
Now, the specific heat capacity of water is approximately 4.18 J/g°C.
Plugging everything into the formula, we get:
Energy = (4.00 × 10^11 m³ × 1.00 g/cm³ × 0.001 kg/g) × 4.18 J/g°C × (11.0°C - 9.0°C)
Now, it's math time! *drum roll, please*
(b) Before we dive into part (b), let's make sure we know what we're dealing with. A 1250-MW electric power plant means it produces 1250 million watts of power. So, in one year, it would generate:
Energy = Power × Time
Energy = 1250 × 10^6 W × 1 yr
So, to find how many years it would take to supply the energy required to raise Lake Erie's temperature:
Years = Energy required / Energy supplied by the power plant
Plug in the values and calculate away!
And that, my friend, concludes our swimming adventure through calculations. Was it as refreshing as a dip in the lake? I sure hope so!
To calculate the energy required to raise the temperature of the water in Lake Erie, we can use the formula:
Q = mcΔT
Where:
Q is the energy required,
m is the mass of the water,
c is the specific heat capacity of water, and
ΔT is the change in temperature.
(a) Energy Required:
First, we need to find the mass of the water in Lake Erie. The density of water is approximately 1000 kg/m^3.
Mass = density * volume
Mass = 1000 kg/m^3 * 4.00 * 10^11 m^3
Next, we can calculate the energy required:
Q = mcΔT
Q = (mass) * (specific heat capacity of water) * (change in temperature)
The specific heat capacity of water is approximately 4186 J/(kg°C).
Q = (mass) * 4186 J/(kg°C) * (11.0°C - 9.0°C)
Now, let's calculate the values step by step:
Mass = 1000 kg/m^3 * 4.00 * 10^11 m^3
Mass = 4.00 * 10^14 kg
Q = (mass) * 4186 J/(kg°C) * (11.0°C - 9.0°C)
Q = (4.00 * 10^14 kg) * 4186 J/(kg°C) * 2.0°C
Q = 1.67 * 10^18 J
Therefore, the energy required to raise the temperature of the water in Lake Erie from 9.0°C to 11.0°C is 1.67 * 10^18 J.
(b) Now let's calculate the number of years it would take to supply this amount of energy using a power plant with an exhaust energy of 1250 MW.
Remember, power is the rate of energy transfer, so we can use the formula:
Power = Energy / Time
With rearrangement, we can solve for time:
Time = Energy / Power
Using the given values:
Energy = 1.67 * 10^18 J
Power = 1250 MW
First, convert the power to Joules:
1 MW = 1 * 10^6 W
1 W = 1 J/s
1250 MW = 1250 * 10^6 W = 1250 * 10^6 J/s
Now, let's calculate the time:
Time = Energy / Power
Time = (1.67 * 10^18 J) / (1250 * 10^6 J/s)
Time = 1.336 * 10^12 seconds
To convert seconds to years, divide by the number of seconds in a year:
1 year = 365 days * 24 hours * 60 minutes * 60 seconds
Time in years = (1.336 * 10^12 seconds) / (365 * 24 * 60 * 60 seconds/year)
Time in years = 42.32 years (rounded to two decimal places)
Therefore, it would take approximately 42.32 years to supply the required energy using a power plant with an exhaust energy of 1250 MW.
To calculate the energy required to raise the temperature of a given volume of water, we can use the formula:
Q = m * c * ΔT
Where:
Q is the energy (in joules)
m is the mass of the water (in kilograms)
c is the specific heat capacity of water (in joules per kilogram per degree Celsius)
ΔT is the change in temperature (in degrees Celsius)
To calculate the energy required to raise the temperature of Lake Erie, we need to know the mass of the water. We can calculate the mass using the density of water and the volume of Lake Erie.
The density of water is approximately 1000 kg/m^3.
So, the mass of Lake Erie is:
mass = density * volume
mass = 1000 kg/m^3 * 4.00 * 10^11 m^3
Now, let's calculate the energy required to raise the temperature using the specific heat capacity of water. The specific heat capacity of water is approximately 4186 J/(kg °C).
(a) Energy required:
ΔT = 11.0 °C - 9.0 °C = 2.0 °C
Q = m * c * ΔT
Q = (mass of Lake Erie) * (specific heat capacity of water) * (change in temperature)
Note: Make sure to use the same unit for mass and specific heat capacity (kilograms).
(b) To calculate how many years it would take to supply this amount of energy using the exhaust energy of an electric power plant, we need to convert the power output to energy over a period of time.
The power output of the power plant is 1250 MW (megawatts). To convert this to energy, we need to multiply the power by the time:
Energy = Power * Time
We need to find the time (in years), so we can rearrange the equation:
Time = Energy / Power
Now, we can substitute the values and calculate the time required.