a cylinder of gas is at a room temp (20 c) the air conditioner breaks down and the tempture rises to 58c what is the new pressure of the gas relative to its initial pressure?
p₁/T₁=p₂/T₂
T₁=273+20 =293 K,
T₂=273+58=331 K
p₂=p₁T₂/T₁= ...
1.12
To determine the new pressure of the gas relative to its initial pressure, we can use the ideal gas law equation, which states:
\(P_1 \cdot V_1 / T_1 = P_2 \cdot V_2 / T_2\)
Where:
- \(P_1\) is the initial pressure of the gas
- \(V_1\) is the initial volume of the gas
- \(T_1\) is the initial temperature of the gas
- \(P_2\) is the new pressure of the gas
- \(V_2\) is the final volume of the gas (which remains constant in this case)
- \(T_2\) is the new temperature of the gas
Given:
\(T_1 = 20 \,^{\circ}\mathrm{C} + 273.15 = 293.15\, \mathrm{K}\)
\(T_2 = 58 \,^{\circ}\mathrm{C} + 273.15 = 331.15\, \mathrm{K}\)
Since the volume of the gas is not changing, we can cancel out \(V_1\) and \(V_2\) from the equation, giving us:
\(P_1 / T_1 = P_2 / T_2\)
Rearranging the equation:
\(P_2 = P_1 \cdot T_2 / T_1\)
Now we can calculate the new pressure:
\(P_2 = P_1 \cdot (331.15) / (293.15)\)
To determine the new pressure of the gas when the temperature increases, we can use the ideal gas law equation:
PV = nRT
Where:
P is the pressure of the gas
V is the volume of the gas (assuming it remains constant)
n is the number of moles of gas (assuming it remains constant)
R is the ideal gas constant (atmospheric pressure)
T is the temperature in Kelvin (K)
First, we need to convert the temperatures from Celsius to Kelvin. The conversion formula is:
T(K) = T(C) + 273.15
Initial temperature, T1 = 20°C + 273.15 = 293.15 K
New temperature, T2 = 58°C + 273.15 = 331.15 K
Since the volume and the number of moles remain constant, we can simplify the equation to:
P1/T1 = P2/T2
Now we can plug in the values:
P1/293.15 = P2/331.15
To find the new pressure, P2, we can rearrange the equation:
P2 = (P1 * T2) / T1
Substituting the values:
P2 = (P1 * 331.15) / 293.15
So the new pressure of the gas relative to its initial pressure is P2/P1 = (P1 * 331.15) / (P1 * 293.15) = 331.15 / 293.15
Calculating the actual value:
P2/P1 = 1.128
Therefore, the new pressure of the gas relative to its initial pressure is 1.128.