1000cm3(cude) of air at 20C and 101.35kpa is heated at constant pressure until its volume doubles. (A) use the ideal gas equation to calculate the final temprature of the gas? (B) calculate the work done by the gas as it expands.
P V = n R T
P does not change
T1 = 20 + 273 Kelvin
n does not change
R does not change
so
V/T = constant
V2/T2 = V1/T1
2000/T2 = 1000/293
T2 = 2*293 = 586 Kelvin = 313 Centigrade
(B) constant pressure = 101.35 *10^3 Newtons/meter^2
Force on inner surface = 4 pi r^2 * P
work done for change in radius dr = 4 pi P r^2 dr
work done frrom R1 to R2 = 4 pi P (R2^3/3-R1^3/3)
= (4/3) pi P (R2^3/R1^3)
or in other words work done at constant pressure
= P (V2-V1)
V2 = 2000 cm^3 (1 m/10^2 cm)^3 = 2000/10^6 meter^2
so V2 - V1 = 10^3/10^6 = 10^-3 meter^3
so
work done = 101.35*10^3 * 10^-3 = 101.35 Joules
since the has been done by the gas,why could not the answer is negative?
Selam
A) Well, let's heat things up with some calculations. We can use the ideal gas equation, which is PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature in Kelvin.
First, we need to convert from 20 degrees Celsius to Kelvin. To do that, we add 273.15 to the Celsius temperature. So, 20 degrees Celsius + 273.15 = 293.15 Kelvin.
Now, let's plug in the values we know: P1 = 101.35 kPa, V1 = 1000 cm³, T1 = 293.15 K. We also know that the volume doubles, so V2 = 2 * V1 = 2 * 1000 cm³ = 2000 cm³.
Since the pressure remains constant, we can rewrite the ideal gas equation as (V1/T1) = (V2/T2). Plugging in the values, we get:
(1000 cm³ / 293.15 K) = (2000 cm³ / T2).
To solve for T2, we'll cross-multiply and get T2 = (2000 cm³ * 293.15 K) / 1000 cm³. Doing some math magic, the final temperature (T2) turns out to be approximately 586.3 K.
B) Now, let's calculate the work done by the gas during expansion. The work done (W) can be determined using the equation W = P∆V, where P is the pressure and ∆V is the change in volume.
Since the pressure is constant at 101.35 kPa, and the volume doubles from 1000 cm³ to 2000 cm³, we plug in the values:
W = (101.35 kPa) * (2000 cm³ - 1000 cm³).
Now, we just crunch the numbers: W = 101.35 kPa * 1000 cm³ = 101,350 J (Joules).
So, after all the calculations, we find that the final temperature is approximately 586.3 K, and the work done by the gas as it expands is 101,350 Joules. And there you have it!
To solve these problems, we can use the ideal gas law equation: PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin.
(A) To calculate the final temperature of the gas when the volume doubles, we need to find the initial and final values of pressure, volume, and temperature.
Given:
Initial volume (V1) = 1000 cm³
Initial temperature (T1) = 20°C
Initial pressure (P1) = 101.35 kPa
Final volume (V2) = 2 * V1 = 2 * 1000 cm³ = 2000 cm³ (since volume doubles)
We need to convert all the values to the SI unit system:
Temperature conversion from Celsius to Kelvin:
T1 = 20°C + 273.15 = 293.15 K
Pressure conversion from kilopascals to pascals:
P1 = 101.35 kPa × 1000 = 101,350 Pa
Now, we can use the ideal gas law equation to find the final temperature:
(P1 * V1) / T1 = (P2 * V2) / T2
Plugging in the known values:
(101,350 Pa * 1000 cm³) / 293.15 K = (P2 * 2000 cm³) / T2
Simplifying the equation:
P2 / T2 = (101,350 Pa * 1000 cm³) / (293.15 K * 2000 cm³)
P2 / T2 = 173.5 Pa / K
Since the pressure and volume are constant (due to constant pressure), we can solve for T2:
T2 = (P2 * T1) / P1
T2 = (173.5 Pa * 293.15 K) / 101,350 Pa
Calculating the final temperature:
T2 ≈ 0.5 K
Therefore, the final temperature of the gas would be approximately 0.5 Kelvin.
(B) The work done by the gas (W) as it expands can be calculated using the formula: W = P * ∆V, where P is the pressure and ∆V is the change in volume.
Given:
Initial pressure (P1) = 101.35 kPa
Final volume (V2) = 2 * V1 = 2 * 1000 cm³ = 2000 cm³
Initial volume (V1) = 1000 cm³
Converting the pressure from kilopascals to pascals:
P1 = 101.35 kPa × 1000 = 101,350 Pa
Calculating the change in volume:
∆V = V2 - V1 = 2000 cm³ - 1000 cm³ = 1000 cm³
Now, we can calculate the work done by the gas:
W = P1 * ∆V
W = 101,350 Pa * 1000 cm³
W = 101,350,000 J
Therefore, the work done by the gas as it expands is approximately 101,350,000 Joules.