7 Moles of a gas initially at temperature 300 K is compressed adiabatically from a volume of 1000 cm3 to a volume of 316 cm3. To the nearest tenth of a kJ what is the work done by the piston? (It is a monatomic ideal gas.)
I have W = (3/2)nRT1(V1^(gamma-1) V2^(1-gamma) times -1)
I tried to set it up and I got 56.4 but the answer was 30.3. Can someone tell me what I did wrong and tell me how to properly calculate it? Thanks!
v= 7 moles,
T₁=300 K
V₁=1000cm³= 10⁻³ m³
V₂=0.316•10⁻³ m³
W= {(vRT₁)/(γ-1)} •[1-(V₁/V₂)^(γ-1)]=
γ=(i+2)/I =(3+2)/3 = 5/3
γ-1=5/3 -1 =2/3= 0.667
W=- {(vRT₁)/(γ-1)} •[1-(V₁/V₂)^(γ-1)]=
= - { 7•8.31•300/0.667}•[1- (10⁻³/0.316•10⁻³)^0.667] =
= 30252 J =30.252 kJ ≈ 30.3 kJ
An ice skater spinning with outstretched arms has an angular speed of 5.0\({\rm rad/s}\) . She tucks in her arms, decreasing her moment of inertia by 19\({\rm \%}\) .
By what factor does the skater's kinetic energy change? (Neglect any frictional effects.)
To calculate the work done by the gas during an adiabatic compression, you can use the formula:
W = (gamma / (gamma - 1)) * P1 * V1 * (V2^(1-gamma) - V1^(1-gamma))
In this formula, gamma represents the heat capacity ratio (Cp / Cv) for a monatomic ideal gas, which is approximately 5/3. P1 is the initial pressure, V1 is the initial volume, and V2 is the final volume.
Let's proceed with the calculation using the given values:
n = 7 moles
T1 = 300 K
V1 = 1000 cm^3
V2 = 316 cm^3
gamma = 5/3
First, let's calculate the initial pressure, P1, using the ideal gas law equation:
PV = nRT
P1 * V1 = nRT1
P1 = (nRT1) / V1
Now, substitute the known values:
P1 = (7 * 8.314 J/mol·K * 300 K) / 1000 cm^3
Convert cm^3 to m^3:
1 cm^3 = 1 x 10^(-6) m^3
P1 = (7 * 8.314 J/mol·K * 300 K) / (1000 * 10^(-6) m^3)
Next, calculate the work done using the formula:
W = (5/3 / ((5/3) - 1)) * P1 * V1 * (V2^(1-(5/3)) - V1^(1-(5/3)))
W = (5/3 / (2/3)) * P1 * V1 * (V2^(-2/3) - V1^(-2/3))
Now, substitute the values of P1, V1, and V2:
W = (5/3 / (2/3)) * P1 * V1 * (V2^(-2/3) - V1^(-2/3))
W = (5/3 / (2/3)) * (P1) * (V1) * ((V2^(-2/3)) - (V1^(-2/3)))
Plug in the calculated values of P1:
W = (5/3 / (2/3)) * (((7 * 8.314 J/mol·K * 300 K) / (1000 * 10^(-6) m^3))) * (1000 cm^3) * ((316 cm^3)^(-2/3) - (1000 cm^3)^(-2/3))
Now, convert cm^3 to m^3:
W = (5/3 / (2/3)) * (((7 * 8.314 J/mol·K * 300 K) / (1000 * 10^(-6) m^3))) * (1000 * 10^(-6) m^3) * ((316 * 10^(-6) m^3)^(-2/3) - (1000 * 10^(-6) m^3)^(-2/3))
Simplify and evaluate the expression to get the final answer.