the temperature & the pressure
The pressure in a constant-volume gas thermometer is 0.700 atm at 100 Celsius degree and 0.512 atm at 0 Celsius degree. a) what is the temperature when the pressure is 0.040 atm? b) what is the pressure at 450 Celsius degree?
We use the gas equation PV/T = RT to determine the temperature and pressure at the constant volume. In the formula P stands for pressure , V volume and T for temperature in kelvin which is equal to t in celsius+273.15 kelvin.Thereore,
P1V1 = R T1 gives :
0.7*0 V=R(273.15+100)..........(1) for boiling and
0.512 V= R(273.15)..................(2)
R = 1.875921211*10^(-3)V from (1) and 1.844279*10^(-3)7V from (2) for
To find the temperature when P = 0.040 atm.
PV = RT . We use R value from the 2nd for lower pressure.
0.040V = (1.8442797*10^(-3)V)T. Or
T = 0.040V/[(1.8442797*10^(-3)V] = 21.3398 kelvin = (21.3398-273.15) = -251.8102 deg Celsius.
To find the pressure at 450 C, we use the R from the eq(1).
PV = RT. Or
P = RT/V = [1.875921211*10^(-3)V ]* (273.15+450)/V = 1.356572 atmospheric pressure.
As the mass and volume of the gas in the thermometer is constant, the change in pressure of the gas is directly proportional to the rise of temperature. Stated in other words rise in pressure per degree rise in temperature is constant.
Pressure at 100 degrees Celsius = 0.700 atm
Pressure at 0 degrees Celsius = 0.512 atm
Rise in pressure for 100 degrees rise in temperature = 0.700 0.512 = 0.188 atm
Rise in temperature per degree Celsius rise in temperature = 0.188/100 = 0.00188 atm
Temperature in Celsius corresponding to any pressure p is equal to:
t = (p - p0)/(rise in pressure per degree Celsius temperature)
= (p - 0.512)/0.00188
Therefore temperature corresponding to pressure of 0.040:
= (0.040 - 0.512)/).00188 - 0.112/0.00188 = 59.57 degree Celsius.
Similarly, pressure p in atm at any given temperature t is given by:
p = p0 + t*(Rise in temperature per degree Celsius rise in temperature)
= 0.512 + t*0.00188
Therefore the pressure at 450 degree Celsius:
= 0.512 + 450*0.00188 = 0.512 + 0.846 = 1.358 atm