The internal energy of ideal gases only depends on temperature. When this is used with the energy balance, the following equation can be derived to relate T and V during an ideal gas adiabatic process: T2 T₁ V₂ -()* = R/Cv For real gases the internal energy is a function of temperature and molar volume (or pressure). The equation which describes is: ӘР + [T (OF), -P] JU Plav dV dU = CydT+ using, develop an equation (in terms of T and V) for the adiabatic compression of a gas following the van der Waals EOS.

3. Potential density D is defined as the density of which dry air would attain if it were transformed reversibly and adiabatically from its existing conditions to a standard pressure po (usually 1000 mb). If the density and pressure of the air are p and p, respectively, it can be proven that the potential density can be expressed as, D = P(Pojev/cp P where cp and c, are the specific heats of air at constant pressure and constant volume, respectively. (a). Calculate the potential density of a quantity of air at a pressure of 600 mb and temperature -15 °C. (b) Show that 1 dD D dz == T (T₁-T) d where I'd is the dry adiabatic lapse rate, I the actual lapse rate of the atmosphere, and T the temperature at height z. [hint: cp - cv = R where R is the gas constant for dry air (R=287 Jdeg¹ kg-¹), cp=1004 Jdeg"¹ kg`¹, and cv=717 Jdeg`¹ kg-¹). (c) Show that the criteria for stable, neutral, and unstable conditions in the atmosphere are that the potential density decreases with height, is constant…

A high-altitude balloon contains helium, whose molar mass is 4.00 g/mol. At theballoon’s maximum altitude, its volume is 792 m3 and the outside temperatureand pressure are -53.0 °C and 5.00 kPa, respectively. Assume that the helium in theballoon is in equilibrium with the outside air temperature and pressure.(a) Use the ideal gas law to find the amount of helium in the balloon at maximumaltitude.(b) Assuming no loss of helium, what would be the volume of the balloon when itwas launched from the ground, where the air temperature and pressure are 20.0°C and 101 kPa, respectively?(c) If the volume of the balloon when it was launched was 65.0 m3, how many molesof helium were lost during the balloon’s ascent?

A group of students have injected 26 g of a gas into an evacuated, constant-volume container at50 C and atmospheric pressure. The students want to heat the gas at constant pressure byallowing some of the gas to escape during the heating. What mass of gas (in grams) must bereleased if the temperature is raised to 400 C?

You are scaling up a sparged stirred tank bioreactor from 10 L to 10,000 L. The volumetric O2 transfer coefficient kla can be estimated using the following empirical correlation: P. 0.4 6, kla = k (v3)0.5 N0.5 where k = empirical constant Pg = power requirement in gassed bioreactor (kW) V, = bioreactor volume (m³) Vs = superficial gas exit speed (m²/min) N = rotational speed of the agitator (rad/min) %3D