# Volume change with reaction

#### Maylis

Gold Member
Hello,

I am looking at reactions that change volume during the reaction, for example ammonia synthesis

$$N_{2} + 3H_{2} \rightleftharpoons 2NH_{3}$$

where it is clear that the products are less moles than the reactants. However, I am thinking of a scenario in a plug flow reactor where this is happening. How can it be that the volume is changing? Aren't the gases just going to occupy the space in the PFR, therefore the volume is constant? Same thing really for a batch reactor too, I mean just because a gas reacts to form less moles, it should still have the volume of its vessel, therefore not change??

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#### Chestermiller

Mentor
I'll give you the answer for a batch reactor, and then, maybe you can figure it out for a PFR.

If the number of moles changes, it doesn't have to be the volume that is affected. It can also be the pressure. In a batch reactor, the mass density remains constant, but the molar density changes. This causes a change in the pressure.

Chet

#### Maylis

Gold Member
So again, I suppose the molar density would change just as well in a PFR, so the pressure and/or temperature would change. If the volume of a PFR is a fixed vessel, and should be constant, then all I see is a change in moles to maintain the equality PV = nRT requires either P or T to change. But the lecture/book is actually talking about a change in volume, very specifically. I don't get how the volume of a gas in a fixed container can change, no matter what the reaction is.

#### Chestermiller

Mentor
So again, I suppose the molar density would change just as well in a PFR, so the pressure and/or temperature would change. If the volume of a PFR is a fixed vessel, and should be constant, then all I see is a change in moles to maintain the equality PV = nRT requires either P or T to change. But the lecture/book is actually talking about a change in volume, very specifically. I don't get how the volume of a gas in a fixed container can change, no matter what the reaction is.
Well, a PFR for a reaction like this is very different from the case of a batch reactor. But, we can figure out what is going on in the PFR by letting the mathematics help us.

Suppose we have a PFR with the reaction B + 3C --> 2D, where B, C, and D are gaseous species. We are going to assume that the shear stress at the wall is very small so that the pressure gradient along the tube is very low, in which case the total pressure P is constant at the inlet pressure throughout the reactor. We are also going to assume that the heat of reaction is zero, so that the temperature throughout the reactor is constant at the inlet temperature T. We are going to assume ideal gas behavior, and that B, C, and D comprise the entire flow stream. Let x be the axial position along the reactor, and let v(x) represent the axial velocity a position x. Let A be the cross sectional area of the tube. Let yB(x), yc(x), and yd(x) represent the mole fractions of the three species, and let Mb, Mc, and Md represent the molecular weights of the three species.

For an ideal gas, in terms of the mole fractions, what are the partial pressures of B, C, and D at a given location x along the reactor? What are the molar densities of the three species at a given location x along the reactor? What are the mass densities of the three species at a given location x along the reactor? What is the total mass density $\rho (x)$of the gas at a given location x along the reactor?

TO BE CONTINUED

Chet

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