Achieving bigger popcorn through vacuum pump

I'm tutoring a high-school level student where one of the questions is asking why fitting a vacuum pump to a popcorn maker would cause the popped kernels to get bigger. For those unfamiliar with popcorn :), it's modeled as a sealed hulled with starch and moisture inside. Heat causes the moisture to mix with the starch to form a gel-like substance, and as the moisture turns to vapor/steam, the pressure inside builds up till it causes the hull to break. The gel-like starch gets pushed out and expands till it solidifies.

Reading a paper by the guy who developed this idea, he modeled the expansion of the water vapor inside the kernel as an adiabatic expansion

[itex]P_{Y}V_{0}^{\gamma}=C_0=\text{constant} \\
\text{where} \\
P_{Y}- \text{yield pressure}\\
V_{0}-\text{initial volume of the kernel}\\
\gamma-C_p/C_v
[/itex]

Resulting manipulations show that the final volume of the kernel would depend on the surrounding pressure around the kernel, thus causing the popcorn to be bigger in a vacuum.

My question is regarding the above equation. Would the yield pressure not depend on the difference between the surrounding pressure and the internal pressure of the hull? I believe the rest of the terms in the equation (i.e. excluding pressure) do not depend on the surrounding pressure so is this correct or have I misunderstood something?