Question

We would like to measure the density (p) of an ideal gas. We know the ideal gas law provides p= , where P represents pressure, R is a constant equal to 287.058 J/kg-K and I represents temperature. We use a pressure transducer to take 15 measurements, all under the same conditions. The sample mean and standard deviation obtained from these 15 pressure measurements are 120,300 Pa and 6,600 Pa respectively. The pressure transducer specifications sheet reports an accuracy of 0.6% of the full-scale output (FSO) and a sensitivity error of 0.3% of the FSO. The FSO for this instrument is 180,000 Pa. In addition, we use a thermocouple to measure the gas temperature. We take 20 temperature measurements and obtain a sample mean and standard deviation of 340 K and 8 K, respectively. The accuracy of the thermocouple is 0.25% of the reading and its hysteresis uncertainty is + 2 K. Calculate the density of the ideal gas and its total uncertainty for a 95% probability. (Sol: 1.2325 +0.1578 kg/m3)

Answer 1

Answer: =

Explanation:

=    P / (R * T) P- Pressure, R=287.058, T- temperature

From the given that

Sample mean(pressure) = 120300 Pa

Standard deviation (pressure) = 6600 Pa

Sample mean(temperature) = 340K

Standard deviation(temperature) = 8K

To calculate the Density;

Maximum pressure = Sample mean(pressure) + standard deviation (pressure) = 120300+6600 = 126900 Pa

Minimum pressure = Sample mean (pressure) - standard deviation (pressure)= 120300-6600 = 113700 Pa

Maximum temperature = Sample mean (temperature) + standard deviation (temperature) = 340+8 = 348K

Minimum temperature = Sample mean (temperature) - standerd deviation (temperature) = 340-8 = 332K

So now to calculate the density:

Maximum Density= Pressure (max)/(R*Temperature (min))= 126900/(287.058*332)= 1.331

Minimum density=Pressure(min)/(R*Temperature (max))= 113700/(287.058*348)= 1.138

Average density= (density (max)+ density (min))/2= (1.331+1.138)/2= 1.2345

cheers i hope this helps