How can a student write an essay about why they should be selected to attend a science trip? How can the essay stand out without sounding stuck up?

Great question.  Writing is hard work, and making it truly stand out can be a daunting task.  I can help with some tips.  

One thing to think about when writing this essay is to think about who your audience is.  Who is going to read your essay?  You will want to tailor your essay to that person.  Knowing who you are writing to will definitely affect your word choice.  If you are writing to your science teacher, you can use science terms, but if your reader doesn't know a thing about science, and your topic has to be about science, then your word choice will be different and you will have to spend more time explaining certain things.  By tailoring your topic and word choice to your reader, that person will likely enjoy your essay more.  

Second suggestion.  Proofread.  I have been through so many student essays over the years.  I can tell you without a doubt that essays that are littered with spelling and grammar errors tell me that the student didn't care enough to make it as perfect as possible.  Your essay will stand out simply by being clear, clean, and error free.  To help you with that kind of proofreading, I recommend using ' "essay lab."  Your essay will be gone over by a Golden Pen Educator, and he/she will give you feedback and suggestions on what is good and what things to add or change to make your essay even stronger.  

Another suggestion is regarding your "stuck up" comment.  If you want to avoid sounding pretentious and stuck up, be truthful.  One simple way to do that is to use word choices that you normally use in "real life."  I can always tell when a student is trying to sound overly impressive in writing.  They start using overly flowery and big words that they have never used at any other time.  It feels forced . . . and stuck up. 

Lastly, pick a topic for your essay that you are passionate about.  Your reader will be able to feel that passion through your writing tone.  If you are excited about your topic, you will write in an excited style.  Your reader will be excited too.  If you are bored with your topic, your reader will be bored too. Good luck!